ADAPTIVE EQUALIZATION
by Nam Ho
B.S., University of Colorado, 1986
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of the
Master of Science
Department of Electrical Engineering
1989
To the GRADUATE SCHOOL OF THE UNIVERSITY OF
COLORADO:
Student's name :________________Nam Dai Ho_______________
Degree : M.S. Discipline: Electrical Engineering
Thesis Title :_____________Adaptive Equalization_________
The final copy of this thesis has been examined by the
undersigned, and we find that both the content and the form meet
acceptable presentation standards of scholarly work in the above
discipline.
Chairperson
Date
This thesis for the Master of Science degree by
Nam Dai Ho
has been approved for the
Department of 
Electrical Engineering
by
JoeE. Thomas
Date: Tt'/MmAet
C/' /
Ho, Nam D. (M.S., Electrical Engineering).
Adaptive Equalization.
Thesis directed by Associate Professor Douglas A. Ross.
In adaptive signal processing, one usually encounters
identification or filtering problems.
Consider the performance function J:
3 = E [e]2 = Z [dfflyffl]2
where e(i) is the difference between the output d(i) of the system
(or the plant) and the output y(i) of the model.
If the model is a MA model then J has one and only one
minimum point and its shape looks like a parabola in one
dimension and a bowl in N dimensions (N >2). The objective of
an identification problem (to find the parameter values of the
model so that they are as close to those of the system as possible)
will be achieved simultaneously with the objective of a filtering
problem (to minimize J).
If the shape of J is flat at the minimum point then
Exhaustive Search technique (ES) is probably the best way to cope
with this situation. ES is easy to understand, it offers very fast
V
convergence and gives very accurate estimates.
This thesis will discuss several techniques to solve
identification and filtering problems and ES will be investigated in
detail.
The form and content of this abstract are approved. I
recommend its publication.
Signed
Douglas A. Ross
ACKNOWLEDGEMENTS
Iv would like to express my sincere appreciation to
Dr. Douglas A. Ross for his support and encouragement during
the preparation of this thesis.
Special thanks are given to the student's parents,
brothers, sisters and friends for their support.
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION .................................. 1
II. GENERALIZED INVERSE .......................... 4
21 AR, MA and ARMA models............... 4
22 Formulation of Normal equation........ 7
23 Solving Normal equation.............. 10
23.1 Direct Inversion............. 10
23.2 Gauss elimination............ 11
23.3 Iterative technique.......... 11
23.4 Generalized Inverse.......... 12
24 Computer simulation.................. 15
2 5 Conclusion .......................... 17
EH. RECURSIVE LEAST SQUARES ALGORITHM .... 18
3 1 Introduction ........................ 18
32 Formulation of the problem........... 18
33 Recursive Least Squares algorithm... 20
34 Some properties of RLS............... 21
35 Computer simulation ................. 23
3 6 Conclusion .......................... 31
IV. EXHAUSTIVE SEARCH TECHNIQUE (ES)............ 32
4 1 Introduction ........................ 32
42 Exhaustive Search technique.......... 34
43 Computation.......................... 44
Vll
a) Exhaustive Search................ 44
b) Recursive Least Squares.......... 45
44 Sample size........................... 48
45 Stepsize.............................. 49
46 Input and output...................... 50
47 Noisy MA process.......................... 51
48 Over/Underestimate the order of the system.... 58
a) Overestimate the order of the system. 62
b) Underestimate the order of the system .... 62
49 Exhaustive Search for infinite impulse
response (HR) filter................ 64
410 Conclusion........................... 66
V. DISCUSSION AND CONCLUSION.................. 68
BIBLIOGRAPHY.................................. 70
APPENDIX
A. THE PARAMETER VALUES OF THE WEIGHT
VECTOR OF THE SYSTEM, THE ESTIMATES OF
THE WEIGHT VECTOR OF THE MODEL, SUM OF
RESIDUALS R, R.M.S OF R ... OF THE GRAPHS
IN CHAPTERS H, III, IV........................ 72
B. PROGRAM LISTINGS .......................... 85
TABLE OF SYMBOLS
RLS = recursive least squares
ES = exhaustive search
D = number of data points
I = number of iterations
L number of parameters
S/N signal to noise ratio
AS = stepsize
R sum of residuals
TABLE OF FIGURES
FIGURE
21.1 Stochasic model................................... 5
2 2.1 Schematic diagram for identification or filtering problems 7
3 5.1 RLS D=150 1=150 L=6 S/N=infinite
parameter values vs number of iterations........ 25
35.2 RLS D=1000 1=1000 L=6 S/N=infinite
R & log10R vs number of iterations.............. 26
35.3 RLS D=1000 1=1000 L=6 S/N=10
parameter values vs number of iterations........ 27
35.4 RLS D=1000 1=1000 L=6 S/N=103
parameter values vs number of iterations........ 28
35.5 RLS D=1000 1=1000 L=6 S/N=ltf
parameter values vs number of iterations........ 29
3 5.6 RLS D=1000 1=1000 L=6 S/N=10,103,105
R & log10R vs number of iterations.............. 30
4 1.1 The "parabola" shape of the performance function J in one
dimension................................................ 33
42 Schematic diagram of exhaustive search....... 35
42.1 ES D =50 1=10 L=6 S/N=infinite
AS = .1 parameter values vs number of iterations 40
42.2 ES D =50 1=10 L=6 S/N=infinite
AS = .1 R & log10R vs number of iterations...... 41
X
42.3 ES D =50 1=10 L=ll S/N=infinite
AS = .1 parameter values vs number of iterations 42
42.4 ES D =50 1=10 L=ll S/N=infinite
AS = .1 R & log10R vs number of iterations 43
47.1 ES D =50 1=10 L=6 S/N=10
AS = .1 parameter values vs number of iterations 54
47.2 ES D =50 1=10 L=6 S/N=103
AS = .1 parameter values vs number of iterations 55
47.3 ES D =50 1=10 L=6 S/N^O5
AS = .1 parameter values vs number of iterations 56
47.4 ES D=50 1=10 L=6 S/N=10,103,105
R & log10R vs number of iterations 57
CHAPTER I
INTRODUCTION
The field of recursive identification is often viewed as a long
and confusing list of methods and tricks. Methods and algorithms
have been developed in different areas with different applications.
The term "recursive identification" is taken from the control
literature. In statistical literature the field is usually called "sequential
parameter estimation," and in signal processing the methods are
known as "adaptive algorithms."
Systems consist of a wide range of objects whose behavior we
are interested in studying, affecting or controlling. The typical tasks
are related to the study and use of the systems: control, signal
processing (filter designing) and prediction.
Many techniques have been developed in control theory,
communications, signal processing and statistics for solving the
problems involving these tasks.
In order to use these techniques, some knowledge of the
properties of a system i.e., model, must be known.
The model of a system can be represented in one of several
different forms. For examples, graphics models ( properties of the
systems summarized in graphics or in tables), mathematical models
( mathematical relationships between certain variables of a system)
are very important and necessary when complex design problems are
involved.
2
In this thesis, mathematical models: moving average (MA)
and autoregressive moving average (ARMA) will be used.
Basically, there are two approaches to building a
mathematical model of a given system: the first way is to look at
physical laws and relationships between variables inside the system;
from there the model is constmcted. Unfortunately, this is not always
possible: it is timeconsuming and may lead to an unnecessarily
complex model. In addition, the system's properties may change in an
unpredictable manner. The second way is to use the signal produced
by the system; it consists of two general steps: choose a particular
member of a family of candidate models, then use an algorithm to
estimate the parameters of this candidate.
The fundamental theorem in the decomposition of stationary
time series proposed by Wold [1] says: The most general
decomposition of a stochastic process is a moving average one (MA).
An MA process can also be modeled by AR or ARMA; a finite
order MA process requires an infinite order AR representation. AR
and MA are special cases of ARMA.
The correct choice of AR, MA, ARMA and their orders is
very important. Mismodeling may lead to serious errors. Akaike
information criterion (AIC) and Final prediction error (FPE) are
among the choices for estimating the orders.
Estimating (identifying) the parameters is usually done in two
ways: a batch of data is collected first and used to identify the
parameters; this is called offline or batch processing identification.
Online or real time identification means updating the
estimates of parameters as new data are available.
3
The following is the synopsis of the thesis's content and
order:
chapter II Generalized Inverse
This is an offline technique. The identification problem
which leads to normal equation is formulated. Several methods such
as direct inversion, Gausselimination, and iterative technique are
briefly discussed and finally generalized inverse is investigated.
chapter in Recursive Least Squares (RLS)
Here is an online algorithm used for MA only. RLS
algorithm has been investigated extensively in technical literature and
is mentioned here for the purpose of pointing out its pros and cons
and showing, later on, how exhaustive search solves some of RLS's
problems.
chapter IV Exhaustive Search (ES)
One of the problems associated with RLS occurs when the
autocorrelation matrix R in normal equation RW=P is illconditioned.
This is why ES came into being. ES works quite well for finite
impulse response (FIR) filters but not so for infinite impulse response
(IIR) filters. Number of computations, rate of convergence, etc., of
ES are investigated in detail in this chapter.
chapter V Discussion and Conclusion
CHAPTER n
GENERALIZED INVERSE
21 AR. MA and ARMA models
Many discretetime random processes encountered in practice
are well approximated by a rational transfer function model. The idea
of representation of a discretetime random process by a model is that
a random process consisting of highly correlated observations may be
generated by applying a series of statistically independent shocks,"
white noise, for instance, to a model (discretetime linear filter,
figure 21.1 on next page).
If an input driving sequence x(i) and the output sequence y(i)
that is to model the discretetime random process are related by the
linear difference equation:
q p
y(i) = Z b(k)x(ik) Z a(k)y(ik)
k=0 k=l
Where
a(0), a(l), a(2),...a(p), b(0), b(l),...b(q) are constants,
this general model is termed an autoregressive moving average
(ARMA) model.
The transfer function H(z) between input x(i) and output y(i)
is a rational function:
H(z)
B(z)/A(z)
5
MODEL

White noise
Input sequence
x(i)
Discretetime linear
filter with transfer
function H(z)

Discretetime
random process
Output sequence
y(i)
figure 21.1 : Stochastic model
6
p q
A(z) = X a(k)zk B(z) = X b(k)z'k
k=0 k=0
It is assumed that A(z) has all its zeros within the unit circle
of the z plane so that the filter is stable.
If all the a(k) coefficients except a(0) = 1 vanish for ARMA
parameters then:
q
y(i) = Â£ b(k)x(ik)
k=0
and the discretetime random process y(i) is a moving average of
order q; it is denoted as MA(q) process. The model is called MA or
allzero model.
Likewise, if all the b(k) coefficients except b(0) are zero in
ARMA model then:
p
y(i) =  X a(k)y(ik) + b(0)x(i)
k=l
The output sequence y(i) is now an autoregressive process of
order p; it is denoted as AR(p) process and the model is an AR or
all pole model.
Note that MA model is often known as finite impulse response
(FIR) filter while AR, ARMA models are usually known as infinite
impulse response (HR) filters [2], [31.
22 Formulation of Normal equation
Consider the following figure:
d(i)y(i)
figure 22.1 : Schematic diagram for identification
or filtering problems.
Model: one can use AR, MA, ARMA, etc. In this case
MA model (or FIR filter) is used.
x(i) : input
d(i) : desired output or output of the system.
y(i) : filtered output or output of the model.
Assume the system itself is a FIR filter. The impulse
response vectors of the system and the model are:
8
wL* II Wj* ... w ^Ll
WL = [ w0 Wj ... w VVL1
respectively.
In identification problems, the objective is to obtain WL so
that WL is as close to WL* as possible. In filtering problems,
minimizing the sum of squared errors is the goal.
The objective now is to find the "best WL such that the sum
of squared difference between d(i) and y(i):
k k
J = X [e(i)]2 = X [d(i)y(i)]2
i=Ll i=Ll
is as small as possible. J is called the performance function (criterion
function).
For a given sequence of input vectors {X(i)} and scalars
{d(i)}, J is a function of WL only and therefore is a measure of how
well WL performs to produce y(i) which matches d(i). The choice of
WL minimizing J is the value that has the best performance.
Let the input vector be:
XL(i) = [ x(i) x(il) ... x(iL+l) ]T
and the given data be:
x(0), x(l), ... x(k), d(Ll), d(L), ... d(k),
then
9
J = 2[e(i)]2 = 2 [d(i)y(i)]2
i=Ll i=Ll
k k k
J = 2[d(i)]2 WJ 2 XL(i)d(i)  [2 d(i)XLT(i)]WL
i=Ll i=Ll i=Ll
k
+ Wlt [2 XL(i) XJ(i) ]WL
i=Ll
note: y(i) = XJ(i)WL = WLTXL(i)
Define the autocorrelation matrix R and the correlation matrix P as
following:
k k
R 2xl0)xlt P = 2 d(i)XL(i)
i=Ll i=Ll
expand R and P out and write in more explicit form:
k k k
Z x(i)x(i) E x(i)x(ii) E x(i)x(iL+i)
i=Ll i=Ll i=Ll
k1 k1
R = E x(i)x(i) E x(i)x(iL+2)
i=L2 i=L2
' kL+1 *
E x(i)x(i)
i=0
10
p
k
2 d(i)x(i)
i=Ll
k
2 d(i)x(il)
i=Ll
k
2 d(i)x(iL+l)
i=Ll
k
J = X [d(i)]2 WTP PTW + WtRW
i=Ll
k
J = X [d(i)]2 2WtP + WtRW Where W = WL
i=Ll
To minimize J, take the derivative of J with respect to W and set it
equal to zero:
dJ/dW = 2P + 2RW = 0
RW = P (22.1)
(22.1) is called the normal equation.
23 Solving the normal equation
23.1 Direct inversion
The most straightforward approach to solving 22.1 is the.
11
direct inversion of R followed by its multiplication with P. The
inverse of R is defined as:
R1 = adjoint R/det R
While this formulation is fine for theoretical work, it is
hardly adequate for numerical work. As the order of R increases, the
number of computations of R'1 gets out of hand; the limitations on
word length of the machine used and the numerical accuracy of
computational methods also introduce roundoff errors which become
significant if the number of computations is large.
23.2 Gausselimination
Gausselimination (or sucessive elimination) consists of
reducing the system of L equations in L unknowns (for example ) to a
system of (L 1) equations in (Ll) unknowns by using one of the
equations to eliminate one of the unknowns. This process repeats until
only one equation and one unknown are obtained. Once this last
unknown is found, the remaining (Ll) unknowns are calculated by
back substitution.
If matrix R does not have full rank, R'1 does not exist, direct
inversion fails and so does Gausselimination.
23.3 Iterative approximation
Given autocorrelation matrix R and correlation matrix P, the
estimate of WL at (k+1) iteration may be computed according to
this scheme:
WL(k+l) =[ W0(k+1) W! (k+1) ... Wl., (k+1) ]T
WL(k+l) = [IXR], WL(k) + nP ; WL(0) = 0
x : small positive number
12
I: identity matrix
It has been shown in [4] that if i is chosen so that:
lim [ IjiR]k>0
k> infinity
then lim WL(k) >WL*
k> infinity
This scheme is not the only way. Others are steepestdescent, Newton
methods and their modifications.
23.4 Generalized Inverse
Consider again equation (22.1):
RW = P
R is a linear transformation with domain and range spaces in
matrix theory. By introducing arbitrary orthomormal bases for both
domain and range, a simple representation (i.e., diagonal matrix)
can be found for R. Let:
V tvl V2...VJ
u = [U1 u2 ...UJ
be new orthomormal sets of vectors used as bases for both domain
and range respectively. The new coordinates W of W with respect to
V:
W = VHW /. W = VW
In range space the new P' of P :
P = UHP .. P = UP'
p = UHP = UhRW = UhRVW' = R'W'
13
In the new system, R becomes R' with:
R' uhrv R UR'Vh
where "d o
R' 0 0
_
D diagtoj o2 ... oK] , k by k matrix , k < L
UHU I , VHV = I
rhr V(R,hR)Vh RhR Vi = Ci2Vi
rrh U(RR,h) uh RRHUi = i2ui
* (jj2 : eigenvalues of RHR and RRH. vj and are eigenvectors of
RHRandRRH
* The positive square root of at2 i = 1,2,3 ... k, are singular values
of R.
* U and V are not unique because they consist of uj, Vj not uniquely
determined.
To solve RW = P in the least squares sense means to find W
so that IIRWPII2 is minimum.
II RWPII2 = II URVhWP ll2 = II URVhWUUhP ii2
Let Y = VHW and P' = UHP
The problem now is to minimize IIR'YP'II2
Y = [y, y2 ... yL]T
14
P' = [pj p2... pL']T
IIRTFII, = [(aiyiPl02+((y2y2p2')2 ... + (c^p/)2]1'2
i = 1, 2 ... L
Consider 2 cases:
Case I
If R is nonsingular then
By letting
 0 xr Pi'/
and W = VY
Case II
If R is singular, at least one of a. must be zero. For instance
i = k+l, k+2... L, then
IIRYP'llj = ... + (a^jPi')
+ pk+1,2 + ... Pl'2]1/2
let
yi= Pi'/c* for i = 1,2,...k and yt= 0 for i>k
.*. W = VY
Whether R is singular or not, the solution W always exists.
If one lets
R* = diag [ctj1 ct2_1 ... aK1] l
W = VY = VR*P' = VR*UhP = R+P
R+ = VR*Uh is called the generalized inverse of R.
15
There are several algorithms to calculate R+, and they fall
into two broad groups. The first group consists of full rank
factorization and singular value decomposition (SVD); the iterative
techniques are in the second group.
One of the commonly discussed iterative techniques in the
second group is due to BenIsrael [5], this and its variants are not
competitive for large matrices with SVD.
The reader interested in the first group is referred to [6] for
full rank factorization and to [7] for details on SVD. This group
could be divided into two smaller ones. Full rank factorization and
other methods based on some variations of Gausselimination, call
these the "elimination methods" group. The second one consists of
SVD only.
If one suspects R is severly illconditioned and /or very high
accuracy is needed, the best idea of all is to compute in exact
arithmetic. This will provide R exactly if R has rational entries.
Elimination methods are preferable with exact methods.
24 Computer simulation
The program used in this chapter is written in BASIC
language. A major part of the program: the subroutine Complex
singular value decomposition was developed by P.A Businger and
G.H. Golub fCommunications of ACM Vol. 12, pp. 564565,
October 1696) in FORTRAN and was translated into BASIC.
The input x(i) is generated as following:
x(i) = 2*(RND.5)*SQR(3*.l)
where is multiplication, SQR() = ()1/2 and RND is a function
which generates random numbers between 0 and 1.
16
x(i) is white noise with mean = 0 and variance = .1.
The desired output d(i) is generated by applying x(i) to a MA model.
The relationship between d(i) and x(i) is :
d(i) = g(0)x(i) + g(l)x(il) + ... + g(p)x(ip).
Where g(0) = .5, g(l) = 1, g(2) = .8 ...
See the program Singular Value Decomposition in appendix
B for values of g(i), i = 0,1, 2,... 20. These are the actual values of
the weight vector WL* of the system.
The results in appendix A are the actual values of g(0),
g(l),... g(p) and their estimates for the cases : p = 6,11, 20.
Given input sequence x(i) and desired output sequence d(i),
the program computes the autocorrelation matrix R and the
correlation matrix P in the normal equation RW = P, then the
generalized inverse of R (R+) is calculated, and finally the solution W
is:
W= R+P
Several observations should be pointed out:
* The estimates are fairly accurate even when the number of
data points used is small.
* As the number of parameters (p) increases, so do the
number of data to ensure accuracy.
* The values of singular values ci of R are machine
dependent. Numeric values can be either integer, singleprecision or
doubleprecision. Singleprecision numeric values are stored with 7
digits (although only 6 may be accurate) and doubleprecision
numeric values are stored with 17 digits of precision and printed
in as many as 16 digits.
17
Therefore, if one uses singleprecision and one of the singular
values of matrix R happens to be less than 108 then concluding R is
singular is wrong. This could happen if doubleprecision is used and
one of the singular values is less than 10'17.
Assume that noise e(i) = 2*(RND.5)*SQR(3*D) is added to
the desired output d(i). The signal to noise ratio is defined as:
Signal/noise = (variance of signal)/(variance of noise)
where variance of signal d(i) = .29 ( computed by using 100,000 data
points ) and variance of noise e(i) is D By varying D, different
signal to noise ratios will be obtained.
The estimates are quite good when signal to noise ratio is
high as expected. The data when signal/noise = 10, 103, 105 are in
appendix A.
25 Conclusion
This chapter summarizes several various techniques to solve
least squares problems (RW=P), or equivalently to find the best
impulse response vector W of FIR filter so that sum of squared
errors are minimized.
Direct inversion, Gausselimination, iterative techniques and
generalized inverse are briefly discussed.
When matrix R in the normal equation RW=P is singular,
only iterative and generalized inverse methods are applicable.
There are several algorithms to compute generalized inverse
of a matrix. SVD is one of them and is widely used, "elimination
methods" are recommended when a matrix is illconditioned and /or
high accuracy is needed.
CHAPTER m
RECURSIVE LEAST SQUARES (RLS)
31 Introduction
There are situations in which as new data are available, the
impulse response vector W = WL needs to be updated accordingly
(adaptive filters), but this becomes frustrated and timeconsuming if
one uses batch processing methods discussed in chapter II since R'1
(direct inversion) or R+ (generalized inverse) ... has to be
recalculated. This is one of the motivations for introducing recursive
least squares algorithm (RLS).
32 Formulation of the problem
The formulation of the problem for RLS is proceeded exactly
in the same way in chapter II. Recall equation (22.1)
redefined:
RW = P
kl
R = Â£ X(i)XT(i) = Rk
i=Ll
kl
P = X X(i)d(i) = PK
i=Ll
Assume new data x(k) and d(k) now are available. How
should one update W recursively ?
19
k
^K+l X X(i)XT(i) i=Ll = Rk+ X(k)XT(k)
k
^K+l = S xd(i) i=Ll = PK + X(k)d(k)
WK+1 = R ip ^K+l rK+l
Employing the well known matrix inversion lemma,
sometimes called "ABCD lemma":
(A + BCD)'1 = A1 A1 B [ D A'1 B + C'111 D A1
This identity is proved by multiplying the left with the right
side and getting identity matrix I.
Applying "ABCD lemma" for RK+1 with the following
associations:
A = RK B = X(k) C = 1 D = XT(k)
Rki X(k)XT(k)RKi
1+ XT(k) Rki X(k)
Then
^k+i"1
20
Note
WK = Rk_1 Pjc ( y(k) = XT(k) WK
Let
Zj, = Rk! X(k) XT(k) = qk (scalar)
e(k) = d(k) y(k)
v = 1/(1+ qk) v = vZk
After some algebraic manipulation and simplification:
Wt+I = WK + e(k)ZK*
33 Recursive least squares algorithm fRLSh
RLS consists of the following steps [8]:
1) Accept new data x(k), d(k).
2) Form X(k) by shifting x(k) into the information vector.
3) Compute the prior output (estimated output)
y(k) = XT(k) wK
4) Compute the error e(k)
e(k)
d(k) y(k)
21
5) Compute the filtered information vector ZK
Zk Rki X(k)
6) Compute the scalar qk
% XT(k) zK
7) Compute the gain constant
v = 1/(1+ qk)
8) Compute the normalized filtered information vector
Zk* VZK
9) Update the weight vector
W = vvk+l WK + e(k) ZK*
10) Update the inverse of autocorrelation matrix
V = V w
34 Some properties of RLS
1) To use RLS, first of all, the autocorrelation matrix R and
the weight vector W must be initialized.
For the weight vector, one could choose:
W(0) = 0
For the autocorrelation matrix Rk, the initial value R0 can be
computed in this way :
22
k
R
K+l
Z X(i)XT(i)
i=Ll
k
R.
K+l
Z X(i)XT(i) + 81
i=L2
81
with
I = identity matrix
8 = small positive constant
/. R0i = S1I
2) The introduction of "forgetting factor" p into RLS (0 <
p < 1) will put more weight on the recent data and less weight on
the older ones. The motivation of using p comes from the cases
where the data {X,d} changes its charateristic within record of
data points (nonstationary process). The RLS in this case will be [9] :
ZkV
(1/p) [rki 
]
P+%
P,
k+l
p [ (1/p) d(k)X(k) + pk ]
23
W,
k+l
d(k)y(k)
Wk +  zk
P+%
p = 0 then every data is weighted equally.
3) From [10] one of the statistical properties of RLS is:
8 *
b(k) =  R*!W
k
k : number of iterations
R*1 : inverse of R
W W : weight vectors of the system and the model respectively.
b(k) = E[W(k)] W = the bias
b(k) becomes smaller as k increases. When k approaches infinity then
b(k) approaches zero. 8 should be small to reduce the bias.
35 Computer simulation
From figure 22.1 page 7, let:
WL* = [w0* Wl* WlV]
= the impulse response vector of the system
WL [w0 W]l wL4 ]
= the impulse response vector of the model
L = length of both WL and WL*
24
D = number of data points or number of iterations
R = sum of residuals
The desired output d(i) is generated by applying the input
x(i) = 2*(RND.5)*SQR(3*.l)
to the system The relationship between d(i) and x(i) is:
d(i) = w0*x(i) + w1*x(il) + ... + wL_j* x(iL+l)
where
w0 = g(0) = .5 Wj = g(l) = 1, w2 = g(2) = .8 ...
See the program "recursive least squares" in appendix B for
the values of g(i), i = 0, l,... 5
RLS is used in this chapter to find WL so that WL gets as
close to WL* as possible.
Figure 35.1 (page 25) shows the values of the estimates (w0,
wi wli ft vector W) and the values of the parameters ( w0*,
Wj*... wL_j* of the vector WL*). The dotted lines are the estimates
and the solid lines are the parameters. This is the case where L = 6
and D = 150.
Figure 35.2 (page 26) is the plot of sum of residuals R vs
number of iterations D ( D = 1000).
Figures 35.3, 35.4, 35.5 (pages 27, 28, 29) show the
estimates and the parameters when noise is added to the desired
output. d(i) and the signal to noise ratio S/N = 10,103,105 L = 6,
D = 1000.
25
parameter values
figure 35.1 :
# of data points
# of iterations
recursive least squares
= 150 : # of parameters = 6
= 150 : signal/noise = infinite
26
sum of residuals R
10E 1 1 ]
I
10E 4 4
IK11 11
IK14 14
IK 21 21
It LOGOI)
\.
1
250 500 750 1000
number of iterations
figure 35.2 : recursive least squares
# of data points = 1000 : # of parameters = 6
# of iterations = 1000 : signal/noise = infinite
R = sum of residuals
27
parameter values
figure 35.3 :
# of data points
# of iterations
recursive least squares
= 1000 : # of parameters = 6
= 1000 : signal/noise = 10
28
parameter values
V
L
mmgmm1000
f
.55
1.1
number of iterations
figure 35.4 :
# of data points
# of iterations
recursive least squares
= 1000 : # of parameters = 6
= 1000 : signal/noise = 103
29
parameter values
l.i v__
v. . _
.55
; 2W 400 600 number of iterations 000 1000
.55 i i t
1.1
figure 35.5 :
# of data points
# of iterations
recursive least squares
= 1000 : #of parameters = 6
= 1000 : signal/noise = 105
30
sum of residuals R
figure 35.6 :
# of data points
# of iterations
recursive least squares
= 1000 : # of parameters =
= 1000
6
signal/noise = 10,103,105
: R = sum of residuals
31
Figure 35.6 (page 30) is the plot of R vs D ( D = 1000)
for S/N = 10,103,105.
The data for these graphs are in appendix A.
36 Conclusion
RLS algorithm works quite well when no noise is involved,
gives fairly good estimates but needs many iterations. "Fast" RLS is
one of the modifications of RLS to reduce the amount of
computation.
CHAPTER IV
EXHAUSTIVE SEARCH TECHNIQUE
41 Introduction
Recall the performance function J in chapter II and HI:
k k
J = Z[e(i)]2 = Â£ [d(i)y(i)]2
i=Ll i=Ll
J is the function of weight vector W only, its shape looks
similar to a "parabola" in one dimensional problem (W = [ wQ]) and a
"bowl" in L dimensions (W = [ wQ Wj wL1 ]). Figure 41.1 (next
page) is the "parabola" shape of J in one dimension, the minimum
point of J corresponds to the best impulse response vector W =
WopdmaI = W*
Recall the normal equation:
RW = P
In one dimension, the condition of the matrix R determines
the shape of the "parabola" at the minimum point. If R has full rank
and well/illconditioned, then the "parabola" is sharp/flat at this point.
Generalized inverse and Exhaustive search (ES) are probably the
most suitable ways to cope with R being illconditioned. ES is quite
simple and easy to understand.
33
figure 41.1: The "parabola" shape of the performance
function J vs W in one dimension. W* is the best impulse
response vector.
34
Although the purpose of RLS and ES is to find the optimal
value of the weight vector W (W optimal), their approaches are
different. RLS bases on the concept of updating vector W(kl) by just
the right amount to generate W(k). The idea behind ES is to search
the best impulse response vector Woptimal on the performance function
J. The following sections will examine exhaustive search technique
and its properties.
42 Exhaustive search technique fESl
There are several ways to search Woptimal on the performance
function J: GaussNewton, gradient (results in least mean squares
algorithm), and exhaustive search (ES) [11].
For simplicity, consider one dimensional case (figure 42 on
next page). Let W = [ Wj] and the optimal weight vector Woptimal =
W* = [ Wj*]. Assume one starts with some initial value of Wp i.e.,
W^O). From Wj(0) how should one proceed to the new value Wx(l)
in such a way that W^l) gets closer to Wx*.
Let the value of J at W^O) be Rj and W^O) = Sj then:
Rj = J[Wj(0) ] = J(Sj)
By increasing St by an amount AS, two possibilities can happen:
J(Sj) > J (Sj + AS) or J(Sj) < J (Sj + AS)
a) If J(Sj) > J (Sj + AS), let:
Rx = J(Sj) and R2 = J (Sj + AS) = J (S2)
35
figure 42: Schematic diagram of exhaustive
search technique in one dimension
36
This situation corresponds with J [W^O)] being on the left
side of the parabola representing the performance function J.
Therefore, increasing Sx will make J decrease, it is the hint giving one
which direction to go. Next, increase S2= Sj + AS an amount AS,
namely:
S3 = S2 + AS
If J (S3) < J (S2) then Rx = J (S2) and R2 = J (S3) and
increase S3, i.e., S4 = S3 + AS evaluate R3 = J (S4). Repeat this
procedure until Rt > R2 and R3 > R2. Fitting a parabola R through
these three points Rp R2 and R3:
R = a + bS + cS2
Three unknowns a, b, c are determined if Rp R2 and R3 are known.
Take derivative of R with respect to S and set it equal to 0.
dR b
 = 2cS + b = 0 .*. S . =
min
dS 2c
Smin will be equal to the new value of W^O):
S . = W.(l)
min lv '
Note that the parabola fitting through Rp R2, R3 and the one
representing J are not the same and neither are Smin and Wj*.
37
What is the value of in term of Rlt ^, R^, Sv S2, S3?
^min f( Rp ^2> R3) Sj, S2 S3)
AS = S2Sj = S3S2
S2 = Si + AS ) S3 " S2 + AS = Sj + 2AS
R = a + bS + cS2
Ri = a + bSj + CSX2 (S15 Rj) (1)
*2 = a + bS2 + CS22 (S2, Ra) (2)
3 a + bS3 + CS32 (S3, R3) (3)
Use Cramer's rule to find b, c from (1), (2), (3) and S = (b/2c)
1 R[ S2
1 R, S?
1 R, Sj2
Si S2
S, Sf
S3 S,2
b
1
1
1
38
b
c
c
min
 AS [ (R3 iy (Sj+ s2) (Rj Rj) (S3+ s2) ]
1 S, S,2
1 S32
1 S3 S32
1 s, *1
1 s>
1 S3 *3
1 s, S,2
1 83 Â¥
1 s, Â¥
AS [ Rr R2+ R3 ]
1 S, . S2
1 S2 S22
1 S3 S32
0.5 [ (R3 R2) (S1+ s2)  (R2 Rj) (S3+ s2) ]
Rr R2+ R3
Repeat the above procedure until:
Wj(0)> Wj(l)> Wj(2)>... Wj*
39
b) If J(S{) < J (Sj + AS), let:
Rj = J (Sj + AS) = J (S2) and R2 = J (Sj).
In this case J [ W^O) ] is on the right side of the parabola
representing the performance function J. Increasing S will make J
increase, going in other direction should decrease J.
Next step is to decrease S1 an amount of AS, S3 = SjAS,
(AS>0) and let R3 = J (S3). From here on, the procedure is
proceeded exactly the same as in case a).
ES can be easily extended to L dimensional problem where:
W = [W0 Wx ... WL.J
In this case each Wj, i= 0, 1, 2,... Ll is used to minimize J
one at a time (WQ is first, Wx is next,... and finally WL1). After this
the process is repeated again and again ... until J can not be reduced
further or Woptimal is achieved.
Figure 42.1 (page 40) shows the estimates (dotted lines) of
the weight vector of the model and the parameters (solid lines) of the
weight vector of the system vs number of iterations I. Where:
length of weight vector L = 6
number of data points D = 50
number of iterations I = 10
40
parameter values
figure 42.1 :
# of data points
# of iterations
signal/noise =
exhaustive search
= 50 : # of parameters = 6
= 10 : stepsize = .1
infinite
sum of residuals R
R LOG(R)
number of iterations
figure 42.2 : exhaustive search
# of data points =50 : # of parameters = 6
# of iterations =10 : stepsize = .1
signal/noise = infinite : R = sum of residuals
42
parameter values
figure 42.3 :
# of data points
# of iterations
signal/noise =
exhaustive search
= 50 : # of parameters =11
= 10 : stepsize = .1
infinite
43
sum of residuals R
number of iterations
figure 42.4 : exhaustive search
# of data points =50 : # of parameters = 11
# of iterations =10 : stepsize = .1
signal/noise = infinite : R = sum of residuals
44
Figure 42.2 (page 41): plot of sum of residuals R (or J) vs I.
Where L = 6, D = 50,1 = 10.
Figure 42.3 (page 42) and 42.4 (page 43): the estimates, the
parameter values, R vs I. L= 11, D = 50,1 = 10.
The data of these graphs are in appendix A.
One observation that should be pointed out is that the rate of
convergence of exhaustive search is very fast. After only 5 iterations,
the estimates are very good and the sum of residuals R is in the order
of 109, r.m.s of R is in the order of 105.
The generation of input x(i) and output d(i) was done exactly
in the same way mentioned in chapter in.
43 Computation
This section attempts to compare RLS and ES
computationally.
a) ES
Let
L : number of parameters of the weight vector.
Lg : number of data points used.
mult. : multiplication , add. : addition
sub. : subtraction , div. : division
Each y(i) ( filtered output) needs:
L(mult.) + (Ll) ( add.) > each y(i)
Le [L(mult.) + (Ll) ( add.) ] > Lg y(i)
Each value of R (R1S R2, R3), where:
45
JE
R = (1/ Lg) E [d(i) y(i)]2 requires:
i=l
(Lg 1) (sub.) + Lg (mult.) + (Lg 1) (add.) + l(div.)
+ Lg [L(mult.) + (Ll) ( add.) ] or:
Lg (L+l) number of mult, and
Lg (L+l) 1 number of add. and sub.
Let
A : average number of times when the situation > R2 and R3> R2
occurs.
B : number of iterations for ES ceases to work (optimal values have
been achieved).
The total number of computations for using ES is:
3.(A).(B) [ Lg (L+l)] number of mult.
3.(A).(B) [ Lg (L+l) 1] number of add. and sub.
b) RLS
Going back to page 20 and 21, let:
46
L : number of parameters of the weight vector.
Lj^ number of data points used or number of iterations.
* step 4 and 7 require one sub., one add. and one div.
* steps 3, 6, 8, 9 needs 4L mult, and [(L1)+(L1)+L] = 3L2 add.
* step 5 : L2 mult, and L(L1) add.
* step 10 : L2 mult, and L2 sub.
The total number of computations for one iteration:
[3]+[4L+3L2]+[2L2 +L2+ L2 L]
or
[2L2 +4L]+[L2 + L2 L+3L2+3]
[2L2 +4L] (mult) + [2L2 +2L+1] (add., and sub.)
The total number of calculations for Lg iterations:
Lr [2L2 +4L] number of mult.
Lr[2L2 +2L+1] number of add., sub., div.
As far as number of multiplications is concerned:
RLS : LR[2.L.h+4.L]
ES : Lgt3.AB.LH3.AB]
47
Most of the time LR Lg while AB > L.
Example: assume RLS and ES are used to estimate the parameters of
the weight vector WL* = [ WQ* Wj* with L = 6.
The values of the parameters are:
W0* = .5
W3* = .2
RLS
Wj* = 1
w4* = .4
W2* = 8
W5* = .9
: recursive least squares needs 1,000 iterations in
order to achieve the following results:
w0 = .50000C 1 w, = .9999999 3 to II
W3 = .199999 6 II Â£ .4000007 II
Sum of residua
r.m.s of R
s R = 2.417282E14
= 1.55476IE7
The number of multiplications for using RLS in this case is:
Lr[2.L.L+4.L] = 1000[(2)(6)(6)+4(6)] = 96,000 mult.
ES : i ES needs 7 iterations (B=7) to achieve the above
similar results,the average number of times when the situation R^ R2
48
and R3 > R2 occurs is A = 12 ( actually, A = 82/7). The number of
data points used is Lg = 30, the stepsize AS = 0.1. The following
results are obtained by using ES:
w0 = .5000012 Wj = .9999988 w2 = .7999998
W3 = .1999998 W4 = .3999995 w5 = .8999994
Sum of residuals R = 2.329294E14
r.m.s of R = 1.526203E7
The number of multiplications in this case is:
Le[3.AB.L+3.AB] = 30[3(82)(6)+3(82)] = 51,660 mult.
*) From this example, ES obviously requires less computation than
RLS does.
*) If adaptive stepsize AS is used in ES case, then the number of
computations can be reduced further.
44 Sample size
Box and Jenkins [12] had suggested about 50 observations
(data points) are an adequate sample size and is also the minimum
required number to build a model. A too small sample size results in
poor estimates. While larger one gives more accurate estimates at the
49
cost of more computations. However, computer experiments had
shown 50 data points are more than enough, for the cases of L = 6
and 21 (L : number of parameters) even 20 and 30 data points
respectively are sufficient. There is no rule of thumb how big the
sample size should be, as L increases so does the sample size.
45 Stepsize
The speed of convergence and the accuracy depend very
much on the stepsize AS.
Large AS makes estimates converge fast to optimal values but
the accuracy is sacrified; small AS has opposite effect.
The best approach to compromising these two ( speed of
convergence and the accuracy) is to use adaptive stepsize. In the
beginning of ES, large AS is used to ensure fast convergence, as the
estimates get closer to the optimal values, small AS is used (one can
choose new AS equal to onehalf, onetenth ... of the previous one)
to achieve desired accuracy.
When the matrix R in normal equation RW = P is suspected
to be illconditioned, small AS is highly recommended.
Most of the results obtained in this chapter use AS = 0.1
Again there is no specific rule as to what the proper AS
should be. AS may be as large as 0.1, 1 or as small as 105 10'15 (this
depends on machine being used, single/doubleprecision).
50
46 Input and desired output
It is fairly obvious that some conditions on the input sequence
x(i) must be introduced in order to secure reasonable identification
results. The input must have sufficient rich frequency content to
excite all the modes of the system.
The exhaustive search technique starts with some initial value
of R:
R
R
R
1 k
 Â£[e(i)]2
(kL)
1 k
 Â£ [d y(i)]2
(kD ,=Li
1 k
 Â£[d(i)]2
(kD i=Li
y(i) = 0 i = Ll, L, ... k
This initial value of R depends on absolute value of mean
(Imeanl) and absolute value of variance (Ivariancel) of desired output
signal d(i) which in some cases can be controlled by using proper
input signal x(i) (which is chosen by the user). Small mean and
variance of d(i) are desirable, give small initial value of R.
51
47 Exhaustive search for noisv MA process
One of the problems with ES which limits its utility is ES's
sensitivity to the additive noise ( noise can be added to the input,
system or the output).
Recall the normal equation:
where
RW = P
kl k1
R = ZX(i)XT(i) and P = X X(i)d(i)
i=Ll i=Ll
When noise is added to the input x(i), the autocorrelation
matrix R will change from R to 8R+R. 5R is the result of input noise.
Likewise, when noise is added to the output y(i), the
correlation matrix P will change from P to 8P+P. 8P is the result of
both input noise and output noise.
Then the solution will change from W to 8W+W.
Let
c(R) = IIRII HR'1!! be the condition number
52
where II II denotes any of the operator norms of R: II llx, II ll2...
M = [1 II (8R ) R_1II ]_1.
Let R be nonsingular and W be the solution to normal equation
RW = P.
Then from [13] it had been shown that:
II8WII II8PII HSR 11
 < M. c(R). [ + ] (47.1)
IIWII II Pll IIRII
ft
%
Although (47.1) gives only the upper bound of the
ratio II8WII /lWII, it does explain several important points:.
1) If SR and SP are large then it is reasonable to predict IISWII will
be large.
2) Asume that there is no noise added to the input (which is the case
for computer simulation in this section). Then 8R = 0, and:
M = [1 II (8R) RMl I'1 = 1
and
IISWII IISPII
 < c(R) [] upper bound UB
II Pll
IIWII
53
UB depends on two terms : c(R) and (II8PII / IIPII).
*) Assume II8PII / IIPII is a constant. The term c(R) is the
condition number of matrix R, small or moderate value of c(R)
implies that the equation RW = P is well conditioned. A large c(R),
however, does not imply the equation is illconditioned.
(In numerical analysis, if "small" changes in the data lead to
"large" changes in the solution, the problem is said to be ill
conditioned; otherwise, it is said to be wellconditioned.)
Although a large condition number c(R) does not indicate the
solution W to equation RW = P is illconditioned for every P, it is
true that W is illconditioned for some P.
*) Assume c(R) is a constant .The term II5PII in II5PII / IIPII
corresponds to the additive noise. Large / small noise will lead to
large / small II8PII which will make upper bound UB = c(R) ( IISPII /
IIPII) large / small and therefore there will be large / small change
II8WII. This explains that as more noise is added to the output ( signal
to noise ratio decreases ) the estimates become worse.
So the change 8W in solution W to normal equation RW = P
depends not only on the condition of matrix R (ill/wellconditioned)
but also on the "magnitude" of noise (variance of noise).
For signal/noise ratio =10, the estimates are quite different
from the actual values of parameters ( figure 47.1 on page 54) .
The estimates are getting better ( the difference between the estimates
and the parameters become smaller) as signal/noise ratio increases as
indicated by figures 47.2, 47.3 on pages 55, 56 Figure 47.4 is the
plot of sum of residuals R vs number of iterations I for the cases S/N
= 10,103 and 105.
54
parameter values
1,1
i
0
.55
l.ii
{8It12H181829
number of iterations
figure 47.1 :
# of data points
# of iterations
signal/noise =
10
exhaustive search
= 40 : # of parameters =
= 20 : stepsize = .1
6
55
parameter values
*101214161820
number of iterations
.55
\
1.1
figure 47.2 :
# of data points
# of iterations
signal/noise =
exhaustive search
= 40 : # of parameters = 6
= 20 : stepsize = .1
103
56
parameter values
101214161820
number of iterations
figure 47.3 : exhaustive search
# of data points =40 : # of parameters = 6
# of iterations = 20 : stepsize = .1
signal/noise = 105
57
sum of residuals R
It LOG(lt)
number of iterations
figure 47.4 : exhaustive search
# of data points =40 : # of parameters = 6
# of iterations = 20 : stepsize = .1
signal/noise = 10, 103, 105 : R = sum of residuals
58
48 Overestimate/underestimate the order of the system
Before getting into the details of the consequences one will
get if exhaustive search technique is used for the model having the
order which is greater or less than the order of the system, lets look
at the modeling technique widely used in statistics, namely
BoxJenkins methods.
The BoxJenkins modeling procedure consists of three stages
for finding a model fitting the available data,
stage 1: identification.
Choose one of the models among autoregressive (AR),
moving average (MA), autoregressive moving average (ARMA) ...
models.
stage 2: estimation.
Estimate the parameters of the model chosen at stage 1.
stage 3: diagnostic checking,
check the candidate for adequacy.
If the model is satisfactory at stage 3, then the procedure
stops, otherwise one has to go back to stage 1 or 2 over again until a
"good" model is obtained.
There are quite a few criterions to make sure a model is a
"good" one at stage 3. One criteria is that the model must be able to
predict the future value with good accuracy, the other is that a good
model must contain the smallest number of parameters.
Assume
d(n), d(nl),... d(m) are available data
and a moving average (MA) model of order L is used to fit the above
59
data. The relationship between the weight vector WL* = [ WQ* Wj*...
WL* ], the input white noise x(k) and the filtered output y(k) (output
from MA model) is:
y(k) = W0* x(k) + Wj* x(kl) ... + WL* x(kL)
The objective of BoxJenkins technique is to find WQ*, Wj*...
WL* so that y(k) and d(k) are as well matched as possible with the
smallest value of L.
For the details of how to find WQ* Wj* ... WL* and the
criterions to judge the "goodness" of these values, the interested
readers are referred to [14]. One of the criteria is tvalue (this is an
indication of how significantly one estimate is far from zero). The
following example will clarify the meaning of tvalue).
Example: Take Wa* for instance.
For a given data d(n), d(nl), ... d(m) of some process,
using BoxJenkins technique, Wa* is obtained. Given another set of
data d((n), d(n'l), ... d(m') of the same process (one process has
many realizations. d(n), d(nl), ... d(m); d((n'), d(n'l), ... d(m') are
just two particular ones), another Wa* will be obtained.
Wa' is a random variable, it has normal distribution with
mean = r (for simplicity, let r = 0) and standard deviation S.
At stage 2 (estimation), Wa* is estimated and its tvalue is
found (for example) to be 2. What does this mean? This means Wa* is
significantly different from r = 0. How is this so?
60
A rule of thumb in statistics is that only 5% of the possible
value W would fall two or more times standard deviation S
a
away from r = 0. So t = 2 means t = 2 times standard deviation S,
and therefore Wa* is significantly different from r = 0.
Finally, this means Wa* must be included in the MA model to
ensure the "goodness" of the model. A model missing W* will lead
to bad results such as bad prediction, d(k) and y(k) are not
matched, and large sum of residuals.
If by mistake, Wa* is not in the model, stage 3 will verify this
and one will go back to stage 2 (or 1) to modify the model.
In the BoxJenkins technique, those parameters having small
tvalues will have absolute values close to 0, and they do not
constribute heavily to the value of output y(k) and hence can be
neglected. A parameter is said to be significant in a MA model if it
constributes heavily to the value of the output compared with other
parameters. A small absolute value of a parameter does not mean
that this parameter is not significant. So the term "significant" has
relative meaning here.
Now let's go back to ES.
al Overestimate the order of the system
Let
Ls and LM be the orders of the system and the model respectively.
wj = [W0 Wj* ...WJ ]
weight vector of the system.
61
= [W0 W^.W^]
weight vector of the model.
The desired output d(k) produced by the system and the
filtered output y(k) produced by the model are:
d(k) = W0* x(k) + Wj* x(kl)... + WLs* x(k Ls )
y(k) = W0x(k) + x(kl)... + WLm x(k Lm)
Because all the parameters of WLs* are included in WLm, by using
exhaustive search, one can expect:
w0~~>w0*, W, > w,* ... wLs.......> WJ
and
W W
vvLs +1 vvLs +2
w
Lm
>0
and sum of residuals = X[e(i)]2 = Z[d(i)y(i)]2 is minimized.
Example: (computer simulation)
Let
Ls = 5 : the order of the system
II * [ .5 1 .8 .2 .4 .9]
weight vector of the system.
Lm = 8 : the order of the model.
II 00 Â£ [W0 Wj...W8] weight vector of the model.
The input white noise x(k) and the desired output d(k) are
generated as before in chapter m. The number of data points = 37.
The number of iterations = 10. Using ES, the estimates of the weight
62
vector of the model W8:
W8 = [ .5 .9999999 .7999999 .2 .4 .8999999
6.343689E9 3.395458E8 1.069999E8 ]
sum of residuals = 2.348792E11
d(k) and y(k) are well matched.
b) Underestimate the order of the system
Because not all the coefficients of WLs* are included in WLm,
there will be some coefficients of WLs* left out. By using ES, the
results of its objectives (to minimize sum of residuals, to have the
estimates of the model as close to the parameters of the system as
possible, y(k) and d(k) are well matched) depend not on how many
parameters are excluded but on how significant these excluding
parameters are. The following examples will clarify this:
Example 1:
Ls = 8 : the order of the system .
w8* = [ .5 1 .8 .2 .4 .9 10^ 2. lO'4 3.10'5 ]
weight vector of the system.
= 5 : the order of the model.
w5 = [ w0 wl ...w5 ]
weight vector of the model.
The input white noise x(k) and the desired output d(k) are
generated as before in chapter III. The number of data points = 37.
The number of iterations = 10. Using ES, the estimates of the weight
vector of the model W5 is:
63
W5 = [ .4999712 1.000022 .8000126 .1999798 .399953
.9000398 ]
sum of residuals R = 3.790102E9
In this example, the last three parameters of the system: 10^,
2. 10*4 3.10'5 have very small absolute values, they are much less
significant compared to other parameters. Therefore, excluding these
last three does not lead to bad result at all. In fact, the sum of
residuals is quite small (= 3.790102E9), the estimates are close to
the parameter values and finally d(k) and y(k) are well matched.
Example 2:
Ls = 8 : the order of the system .
W8* = [.5 1 .8 .2 .4 .9 .1 .1 1 ]
weight vector of the system.
Lj^ = 7 : the order of the model.
w7 = [W0 Wj ...w7 ]
weight vector of the model.
The estimates of the weight vector of the model W? :
W7 = [ .3050533 1.158459 .8904336 .1458553
.2210489 .8206268 2.065472E2 9.711713E2]
sum of residuals R = 7.387126E 2
In this example, Wg* has 8 parameters W? has 7 parameters.
Although the model misses only one parameter, the results (sum of
residuals is large the estimates of the model are not close to the
parameters of the system, y(k) and d(k) are not matched) are bad.
64
This is due to the fact that the excluding parameter (1 in this case) is
quite significant. Its absolute value (I II = 1 ) is more or less
equivalent to the absolute values of other parameters).
It is noted that the model with a small number of parameters
is preferred in practice. However, if this model leads to bad results
then one may suspect that some significant coefficients were excluded
and therefore increasing the order of the model may be helpful.
49 ES for Infinite impulse response (TTR'l filters.
Self adjusting or adaptive filters (FIR & IIR) have been
successfully applied to a wide spectrum of problems, ranging from
adaptive control systems, adaptive estimation, to adaptive signal
processing. In a broad sense, most applications in the control and
communication weas can be regarded as "signal processing".
For the most part, signal processing applications have relied
heavily upon well known and well understood adaptive finite impulse
response (FIR) filters. However, in real life, there are common
situations where FIR filters are not practical, and lead to heavy
computation. As the result, in recent years, research has been directed
to extending the adaptive filters to more general, complicated yet
more efficient infinite impulse response (IIR) filters. One of the
advantages of HR over FIR is the substantial decrease in computation.
Exhaustive search has been shown to work successfully for
FIR filter without additive noise or when signal to noise ratio is
high.
In this section, it is attempted to use ES for IIR filters.
ES does not work well for HR as opposed to FIR filters.
Several observations deserve to be pointed out:
*) In filtering problems, the objective is to minimize sum
65
of squared errors Z[e(i)]2= X[d(i)y(i)]2. The estimates of the
weight vector of the model and the parameter values of the weight
vector of the system may be quite different. These differences may be
substantial but are tolerated as long as X[e(i)]2 is minimized.
hi identification problems, the goal is to obtain the estimates
as close to the parameter values as possible.
For FIR filters, the filtering problems and the dentification
problems are achieved simultaneously. This is not so for IIR filters:
minimizing sum of squared errors Z[e(i)]2 does not make the
estimates get close to the parameter values but the inverse is tme.
*) For MA model (FIR filters), the performance
function J = E[e(i)]2 has parabola shape and has one and only one
minimum point. On the other hand, for ARMA model (IIR filters),
the performance function J has several minimum points.
*) To use ES, one must initialize the estimates. Different
sets of initial values can lead to different minimum points and in some
cases may lead to neither minimizing X[e(i)]2 nor good estimates.
Example:
The input white noise x(k) = 2*(RND.5)*SQR(3*1) and the
desired output d(k) produced by the system are related as following:
d(k) = .2x(kl) + .4x(k2) .5x(k3) .9d(kl) + .6d(k2)
Likewise, the relationship between x(k) and the output y(k)
produced by the model is:
66
y(k) = WjxCk1) + W2x(k2) +W3x(k3) +W4y(k1) +W5y(k2).
Using ES with stepsize = .01, the number of data points
used = 50 and the initial values of the estimates are chosen to be:
w, = .18 Wj = .42
w3 = .51 W4 = .89
w5 = .62
After 40 iterations the following results are obtained:
w, = .2115997 w2 = .416828
w3 = .5088732 W4 = .894435
w5 = .5998373
stun of squared errors X[e(i)]2 = X[d(i)y(i)]2 = 1.21141E4
From this example, one can see that although the initial
values are intentionally chosen to be close to the parameter values,
after 40 iterations the estimates do not converge to the parameter
values at all. The sum of the squared errors is not very small. Many
iterations are needed.
410 Conclusion
For MA model (FIR filter), exhaustive search (ES) works
quite well when there is no additive noise. In terms of computation
and accuracy, ES is better than recursive least squares (RLS).
Especially, when the autocorrelation matrix R in normal equation
RW = P is illconditined, ES gives more accurate estimates than RLS
does. This is the main reason why ES came into being.
67
When there is noise involved, none of RLS, ES, generalized
inverse works better than the other two.
For ARMA model, ES does not work well as explained in
section 49.
CHAPTER V
DISCUSSION AND CONCLUSION
The first step in using one of the techniques in chapters 2, 3,
4 is to assume a model. There are several kinds of models, but only
MA and ARM A were used in this thesis.
The second step is to estimate the order of the model using
criterions such as FPE, AIC, CAT...
In identification problems, the objective is to obtain the
estimates as accurately as possible. In filtering problems, one is
concerned about minimizing the difference between the desired
output ( produced by the system or the plant) and the models output
and can tolerate substantial parameter error.
Techniques are classified into two broad groups: offline and
online.
SVD and ES belong to the former. RLS is in the latter
group.
There is great flexibility in selecting computational methods
without any restriction on time, as a result the accuracy of the
estimates is fairly high for offline and preferred by statisticians.
For engineers, both accuracy and time are crucially important
and online techniques are preferred although they have less
accuracy. In addition, in many situations one could not afford to
waste time to collect all the data.
In chapter II, several techniques : direct inversion ,
Gausselimination iterative technique, singular value decomposition
69
were briefly mentioned to solve normal equation RW = P. When R is
singular, all others fail except SVD and iterative techniques.
Chapter III was about recursive least squares algorithm
(RLS). RLS is based on the concept of starting with some initial
values of the weight vector of the model and then using each input
sample (input & desired output) to update the estimates so that the
estimates get closer and finally converge to the parameter values of
the weight vector of the system. RLS works pretty well when there is
no additive noise, the estimates are quite good but requires many
iterations (which means heavy computation).
Chapter IV is about exhaustive search (ES). ES requires less
computation and is more accurate than its counterpart RLS.
Most of the properties of ES in chapter IV were not proved
mathematically but obtained through computer simulation. Computer
simulation is a very important and useful tool for investigating ES
in particular and any other algorithms in signal processing.
Howerver, a serious limitation is that it may not be conclusive. It is
difficult to tell whether a simulation result has universal implication
or merely reflects properties of the chosen data sequence. To obtain
results of more general validity one must use mathematical analysis.
When there is noise involved, none of SVD, RLS ES works
better than the other two. With noise free data ES is the best in term
of accuracy. All the techniques discussed in this thesis are applicable
for MA models but they do not work for ARMA model.
BIBLIOGRAPHY
[1] H. Wold, A study in the analysis of stationary time series,
Almqvist and Wiksell, Uppsala 1938.
[2] Simon Haykin, Adaptive filter theory, PrenticeHall, 1986, pp.
6771.
[3] Steven M. Kay, Modern spectral estimationtheory and
application, PrenticeHall, 1989, p. 109.
[4] Michael G. Larimore, John R. Treichler and C. Richard
Johnson Jr, Theory and design of adaptive filters, John Wiley
& Sons, Inc, 1987, pp. 4344.
[5] A. BenIsrael, A note on an iterative method for generalized
inversions of matrices, Math. Comp. 20, 1966, pp. 439440.
[6] Campbell + Meyer, Generalized inverses of linear
transformation, Pitman publising Limited, 1979, pp. 1619.
[7] see [6], pp 251255.
[8] see [4], pp 91100.
[9] see [4], pp 91100.
[10] see [2], p 392.
[11] D. A. Ross Constrained optimization of correlation data,
private communication, 1989.
[12] Alan Pankratz, Forecasting with univariate Box Jenkins
models John Wiley, 1983, p. 11.
[13] Ben Noble and James W. Daniel, Applied linear algebra,
PrenticeHall, 1977, pp. 170173.
71
[14] see [12]
[15] S. Lawrence Marple, Jr, Digital spectral analysis with
applications, PrenticeHall, 1987, pp. 94104.
[16] see [4], pp. 267270.
APPENDIX A
This appendix contains the parameter values of the weight
vector of the system, the estimates of the weight vector of the model,
sum of residuals R, r.m.s of R ... of the graphs in chapters H, III and
IV.
The programs are in appendix B.
The computer used is Zenith.
73
SINGULAR TALDE DECOMPOSITION
NOHBER OF PARAHETERS : 6
NOHBER 0? Dili POINTS = 10 NOHBER OF DATA POINTS : 20
tbe actoal Valdes THE ESTIHATES THE ACTOAL TALOES THE ESTIHATES
.5 .4999994 .5 .5000004
1 1 1 .9999998
.8 .7999995 .8 .7999998
.2 .2 .2 .1999997
.4 .4000004 .4 .3999995
.9 .9000002 .9 .8999994
NOHBER OF PARAHETERS = 11
HOBBER Of D8Ti POINTS : 10 NOHBER OF DATA POINTS = 20
THE ACTUAL TILDES THE ESTIHATES THE ACTOAL TALOES THE ESTIHATES
.5 .5000003 .5 .5
1 .9999993 1 .9999999
.8 .7999996 .8 .7999991
.2 .2000003 .2 .1999995
A .4000023 .4 .3999993
.9 .9000001 .9 .8999992
.8 .8000005 .8 .7999998
A .4000015 .4 .3999998
.3 .3000021 .3 .3000001
1 .9999984 1 1
.75 .7499993 .75 .7500001
NOHBER OF PARAHETERS = 20
NOHBER Of DATA POINTS 10 NOHBER OF DATA POINTS = 20
THE ACTDAL VALDES THE ESTIHATES THE ACTOAL TALOES THE ESTIHATES
.5 .3176904 .5 .5000009
1 1.041876 1 .9999974
.8 1.058452 .8 .8000009
.2 .8182513 .2 .2000022
.4 .1677476 .4 .4000015
.9 1.44248 .9 .8999991
.8 1.257589 .8 .7999984
.4 .6428237 .4 .3999972
.3 .2983337 .3 .299999
1 1.186358 1 .9999976
.75 6808423 .75 .7500015
1.5 .7937741 1.5 1.5
1.1 .6679647 1.1 1.100001
.6 .1443536 .6 .6000001
1.2 .7956138 1.2 1.200001
.12 5.106133E02 .12 .1200018
1.55 .9926724 1.55 1.550001
1.21 1.109791 1.21 1.209999
.86 .6846545 .66 .6600013
1.9 1.171808 1.9 1.899997
74
SINGULAR 7AL0E DECOMPOSITION
NUMBER OF PARAMETERS = 6 MDMBER OF DATA POINTS = 40
SIGNAL/NOISE : 10 SIGHAL/HOISE = 100
THE ACTUAL VALDES THE ESTIMATES THE ACTOAL VALDES THE ESTIMATES
.5 .496573 .5 .4989162
1 1.035976 1 1.011376
.8 .8325791 .8 .8103024
.2 .1553249 .2 .1858725
.4 .3893638 .4 .3966366
.9 .9987332 .9 .9312219
SIGHAL/HOISE  103 SIGHAL/HOISE = 10*4
THE ACTOAL VALDES THE ESTIMATES THE ACTOAL VALDES THE ESTIMATES
.5 .4996572 .5 .4998915
1 1.003597 1 1.001137
.8 .8032578 .8 .8010301
.2 .1955324 .2 .1985872
A .3989365 .4 .3996637
.9 .9098734 .9 .903122
SIGHAL/HOISE 10*5 SIGHAL/HOISE = 10*6
THE ACTDAL VALDES THE ESTIMATES THE ACTOAL VALDES THE ESTIMATES
.5 .4999657 .5 .4999891
1 1.000359 1 1.000113
.8 .8003258 .8 .800103
.2 .1995532 .2 .1998586
.4 .3998937 .4 .3999664
.9 .9009871 .9 .900312
75
RECURSIVE LEAST SQUARES
ITER
G( 0) G( 1) G( 2) G( 3) G( 4) G( 5)
.5 1 .8 .2 .4 .9
10
.7746811 1.130828 1.128002 3.562172E02 .471651 1.098035
50
.5452403 .9960952 .8340062 .1867677 .4260443 .9096918
100
.5151973 200 .9967139 .8112345 .1996619 .4098652 .9010181
.5037134 .9995578 .8026518 .1991361 .4027318 .9003244
300
.5011802 .9997643 .8008996 .1999186 .4008045 .9000525
400
.5004199 .9999334 .8003286 .1999284 .4003068 .9000047
500
.5001542 .9999701 .8001208 .1999746 .4001105 .9000018
600
.5000513 .9999892 .8000393 .1999934 .4000412 .9000002
700
.5000183 .9999971 .8000151 .1999946 .4000134 .9000005
800
.5000066 .9999981 .8000058 .1999989 .4000034 .8999992
900
.5000021 .9999993 .800002 .1999992 .4000017 .8999996
1000
.500001 .9999999 .8000009 .1999996 .4000007 .8999999
ITER SOM OF RESIDUALS R RMS OF R
10 3.008534E03 5.485011E02
50 3.545921E05 5.954764E03
100 4.051348E06 2.012796E03
200 2.622408E07 5.120946E04
300 2.689385E08 1.639935E04
400 3.586889E09 5.989065E05
500 4.822382E10 2.195992E05
600 5.622787E11 7.498524E06
700 7.407871E12 2.721741E06
800 8.782514E13 9.371507E07
900 1.155176E13 3.398788E07
1000 2.417282E14 1.554761E07
76
RECURSIVE LEAST SQUARES
ITER G( 0) .5 G( 6) .8 SUH OF RESIDUALS R G( 1) G( 2) 1 .8 G( 7) G( 8) .4 .3 RHS OF R G( 3) .2 G( 9) 1 G( 4) .4 G(10) .75 G( 51 .9
10 .3132842 .1138547 .3739014 .7471896 .5B6B756 3.275992 4.25797 .6114748 .6026376 .3636949 1.949431 0 3.516B64
100 .5374023 .7519163 1.339287E04 .9385386 .7251513 .4286757 .3196768 1.157276E02 .148172 .9856912 .399184 .7417548 .95004
200 .50916 .7906353 5.951337E06 .9874006 .7856224 .4069814 .3052762 2.439537E03 .1898938 .9977764 .3907489 .7502281 .9085028
300 .5026838 .7970946 6.182849S07 .9960766 .7952621 .4017215 .3012052 7.86311E04 .1966868 .999275 .3996987 .7496806 .9031326
400 .5011186 .7991148 7.897627E08 .9985184 .7982588 .4005667 .3005791 2.810272E04 .1987953 .9996179 .3997553 .7498695 .9009021
500 .5002956 .7996843 8.681652E09 .9995299 .7994212 .4001393 .3000817 9.317539805 .1995945 .9999771 .3999376 .7500116 .9003593
700 .5000549 .7999479 1.834878E10 .999935 .7999253 .4000371 .3000301 1.354577E05 .1999458 .9999775 .3999893 .7500006 .9000444
800 .5000117 .7999853 1.62634E11 .9999835 .7999765 .4000045 .3000032 4.03279E06 .1999824 .9999984 .3999987 .7500031 .9000178
900 .5000042 .7999946 1.93731812 .9999939 .799992 .4000025 .3000015 1.391873E06 .1999948 .9999996 .3999994 .7500013 .9000054
1000 .5000022 .7999977 3.948236E13 .999997 .7999966 .4000009 .3000009 6.2835E07 .1999977 .9999987 .3999992 .7499995 .9000018
77
RECURSIVE LEAST SQUARES
QF DATA POINTS = 1000
OF ITERATIONS = 1000
I OF PARANETERS = 6
SI6NAL/NQISE RATIO = 10
ITER
S( 0)
.5
10
.7141348
50
.5554639
100
.5628154
200
.56825
300
.4816526
400
.4704792
500
.4895214
600
.4750061
700
.4981464
800
.4886953
900
.5047264
1000
.5382441
ITER
10
50
100
200
300
400
500
600
700
800
900
1000
B( 1)
1
1.623208
1.008998
l.oBsiai
1.013933
1.047915
.9298575
.9352865
.9602142
.9094558
.9992925
.9905889
1.036235
6( 2)
.8
1.253718
.835437
.8333776
.7777376
.8252398
.7594174
.7741495
.8096293
.8344949
.7952107
.8569975
.BQ14445
6( 3)
.3473605
.1796957
.1674355
.2419255
.1660199
.1753052
.1556807
.1676619
.1883573
.1426926
.1890078
.17524B1
RNS OF R
.3218926
.1503082
.1660312
.1617933
.1677168
.171976
.1696752
.1662395
.1705736
.1705327
.168289
.1679637
6( 4)
.4
.8684159
.2713918
.3981438
.3939645
.3861295
.4359998
.3927307
.4140025
.4540443
.4640255
.446578
.4333977
6( 5)
.9
1.321568
.8400678
.8587018
.9182339
.9101685
.9638952
.9340479
.8831844
.9572498
.9530052
.9331438
.9247265
SUN OF RESIDUALS R
.1036149
2.259255E02
2.756636E02
2.617706E02
2.B12B92E02
2.957574E02
2.B7B968E02
2.763557E02
2.909535E02
2.908139E02
2.832117E02
2.821182E02
78
RECURSIVE LEAST SQUARES
I OF DATA POINTS = 1000
I OF ITERATIONS = 1000
I OF PARAMETERS = 6
S1GHAL/N0ISE RATIO = 10*3
ITER
G( 0) G( 1) G( 2) G( 3) G( 4) G( 5)
.5 1 .8 .2 .4 .9
10
.6259846 1.600235 1.456218 .5585368 1.124058 1.375298
50
.5195635 1.044977 .8165253 .1758852 .3916199 .8932839
100
.5130924 1.027508 .8082988 .1858427 .4041152 .8979336
200
.5094396 1.006385 .7986541 .2014733 .4014737 .9012273
300
.4986066 1.006244 .8027991 .195801 .3989227 .9011411
400
.4972594 .9934939 .7960612 .1972776 .4037733 .9063842
500
.4990446 .9937102 .797448 .1954711 .3993407 .9033865
600
.4975370 700 .9960973 .800975 .196721 .4014246 .8983206
.4998269 .9909719 .8034554 .1988193 .4054132 .905725
800
.4988737 .9999376 .7995225 .1942649 .4064055 .9052991
900
.5004745 .9990618 .8057001 .1988983 .4046593 .9033137
1000
.503825 1.003624 .8001443 .1975238 .4033402 .9024724
ITER SUM OF R8SIDUALS R RMS OF R
10 .111524 .3339521
50 4.805839E04 2.192222E02
100 3.496549E04 1.869906E02
200 2.620637E04 .016189
300 2.B19115E04 1.679022E02
400 2.948825E04 1.717214E02
500 2.8764B5E04 .0169602
600 2.763076E04 .0166225
700 2.909183E04 1.705633B02
BOO 2.90B19E04 1.705342E02
900 2.832125804 1.682892802
1000 2.821194B04 1.679641802
RECURSIVE LEAST SQUARES
* OF DATA POINTS = 1000
I OF ITERATIONS = 1000
I OF PARAMETERS = 6
SIGNAL/NQISE RATIO = 1QA5
ITER
S( 0) 6( 1) 6( 2) 6( 3) 6( 4) S( 5!
.5 1 .8 .2 .4 .9
10
.6171702 1.605939 1.476469 .5796545 1.149621 1.380671
50
.5159735 1.048575 .814634 .1755042 .4036427 .8986057
100
.5081202 1.02174 .8057909 .1876835 .4047122 .9018568
200
.5035586 1.00563 .8007459 .1974281 .4022246 .0995269
300
.5005219 1.002077 .8005549 .1987791 .4002021 .9002382
400
.4999376 500 .9998567 .7997253 .1994747 .4005506 .9006331
.4999968 600 .9995522 .7997778 .1994501 .4000018 .9003204
.4997913 .9996855 .0001095 .199627 .4001668 .8998341
700
.4999951 .9991236 .8003518 .1998655 .40055 .9005722
BOO
.4998914 1.000002 .7999538 .1994222 .4006435 .9005283
900
.5000495 .9999096 .0005707 .1998874 .4004673 .9003305
1000
.5003832 1.000363 .0000145 .1997514 .4003345 .9002467
ITER SUN OF RESIDUALS RNS OF R
10 .1138336 .3373924
50 3.753803E04 1.937473E02
100 7.07027E05 8.408489E03
200 7.186342E06 2.680736E03
300 3.24779E06 1.802163E03
400 2.906356E06 1.704804E03
500 2.Q57906E06 1.690534E03
600 2.759286E06 1.661UE03
700 2.905691E06 1.704609E03
800 2.908694E06 1.705489E03
900 2.832199E06 1.6B2914E03
1000 2.821322E06 1.679679E03
80
SIHAOSTIYE SEABCH
f OF PABAHETEBS = 6 OF DATA = 50 STEPSIZB = .1
I TIB G(0) 6(1) 6(2) 6(3) 6(4) 6(5)
.5 1 .5059382 2 .4898913 3 .5007988 4 .500106 5 .5000042 6 .4999999 7 .5 8 .5 9 .4999999 1 .'8 .2 .4 .9
.9435526 .7606144 .2665095 .3501287 .8856738
.9842024 .8083669 .2013362 .396082 .9004196
.9994779 .8007211 .1996734 .3998517 .9000429
1.000037 .8000151 .1999681 .400004 .9000013
1.000005 .7999982 .1999989 .4000009 .8999999
1 .7999998 .2000001 .4 .9
1 .8000001 .2 .4 .9
1 .8 .2 .4 .9
1 .8 .2 .4 .9000001
10 .5 1 .8 .2 .4 .9000001
ITIB SUII OF HESIDOALS B BBS OF fi
1 9.821595E05 9.910396E03
2 3.727433E06 1.930656E03
3 1.227057E08 1.107726104
4 1.098729E10 1.048203E05
5 3.994297E13 6.320045E07
6 9.076767E16 3.012767E08
7 2.815109E16 1.677829E08
8 1.743744116 1.320509E08
9 2.928907E16 1.711405808
10 5.120904E17 7.156049E09
EIHAOSTIYE SEARCH
I OF PARAHETEBS = 11 OF DATA = 50 STEPSIZE = .1
ITER
8(0) 8(1) 8(2)
.5 1 .8
8(6) 6(7) 6(8)
.8 1 .5373538 .4 .3
.7671535 .4518795
.5753536 2 .4582705 .2970887 .274225
.9230392 .7792879
.7671306 3 .4943438 .370235 .3036527
.9846756 .802288
.7969101 4 .5001471 .3961909 .3013491
.9975412 .8012644
.7998453 5 .5003152 .3997685 .3003469
.9997835 .8002174
.8000241 6 .5000717 .4000466 .300061
.999998 .9000235
.8000073 7 .4000138 .3000088
.5000103 1.000004 .8000005
.8000013 8 .5000007 .4000022 .3000009
1.000001 .7999998
.8000001 9 .4000003 .3000001
.4999999 1 .8
.8 10 .4 .3
.5 1 .8
.8 .4 .3
ITEB SDH OF RESIDUALS B
l 2.04834E03
2 9.515579E05
3 3.174369106
4 7.441876E0B
5 1.528297E09
6 4.589051111
7 1.125981E12
8 1.516377114
9 3.453488116
10 3.106543116
6(3) 6(4) 6(5)
.2 .4 .9
6(9) 6(10)
1 .75
.3710172 .2474033 .3677823
.8567359 .8867448
.2486658 .3506067 .9031748
.984106 .7401089
.2061994 .3891598 .9035449
.9980882 .7484755
.2001024 .3986763 .9007799
.9997742 .7497903
.1998897 .3999286 .9001197
.999966 .7499876
.1999727 .4000141 .9000104
.9999961 .7500013
.1999964 .4000042 .9000001
.9999996 .7500006
.1999997 .4000007 .8999999
1 .7500001
.2 .4000001 .9
.9999999 .7500001
.2 .4 .9
1 .75
BUS OF B
.0452586
9.754784K03
1.781676803
2.72798E04
3.909344E05
6.774254E06
1.061123806
1.231413807
1.858356S08
1.762539808
82
EXHAUSTIVE SEARCH
I OF DATA POINTS = 40
I OF ITERATIONS = 10
OF PARAHETERS = 6
SIGNAL TO NOISE RATIO = 10
ITER 6(0) S(l) 6(2) 6(3) 6(4) 6(5)
.5 1 .5791272 1 .8 .2 .4 .9
1.100057 .9108996 5.006526E02 .1510451 .9236923
L .4647143 7 1.078307 .7706338 4.498323E02 .3231962 .9823988
1.020781 .7918397 .1086689 .3681998 1.004547
1.020338 .7943437 .1121429 .3702465 1.005102
0 .5242792 7 .5243801 Q 1.020436 .7950426 .1128936 .3706305 1.005157
1.020503 .7952063 .113036 .3706873 1.005152
Q .5243895 Q 1.020526 .79524 .1130576 .3706916 1.005147
T .5243883 1.020532 .795246 .1130596 .3706906 1.005145
10 .5243871 1.020534 .7952468 .1130592 .3706898 1.005145
ITER 1 1 SUN OF RESIDUALS R RHS OF R STEPSIZE .1
2.655537E02 .1629582
2 1.888654E02 .1374283
3 1.801856E02 .1342332
4 1.795079E02 .1339805
5 1.794738E02 .1339678
6 1.794722E02 .1339672
7 1.794722E02 .1339672
8 1.794721E02 .1339672
9 1.794722E02 .1339672
10 1.794721E02 .1339672
83
EXHAUSTIVE SEARCH
I OF DATA POINTS = 40
I OF ITERATIONS = 10
OF PARAMETERS = 6
SI6NAL TO NOISE RATIO = 10A]
ITER
G(0 G(l) 6(2) 6(3) 6(4) 6(5)
.5 1 .5476146 1 .8 .2 .4 .9
1.067056 .8826056 2.413719E02 .1925286 .8378168
L .447066 3 .4834038 A 1.050989 .7726919 .1272363 .3537032 .8907127
1.006088 .7886431 .1750563 .3862042 .9071142
7 .4987363 5 .5018057 L 1.002076 .7963551 .1873996 .39492 .9100473
1.001835 .7987092 .1905129 .3967018 .9104932
0 .5023555 7 1.001958 .7993452 .1911683 .3970246 .9105294
/ .502436 fl 1.002025 .7994901 .1912885 .397069 .910522
d .5024418 9 .5024401 1.002046 .7995193 .1913056 .3970712 .9105169
1.002052 .7995241 .1913066 .3970699 .910515
10 .5024391 1.002053 .7995248 .1913062 .3970692 .9105146
ITER
ITER SUM OF RESIDUALS R RMS OF R
1 7.26021E03 8.520686E02
2 9.7U315E04 3.U6298E02
3 2.380349E04 1.542838E02
4 1.823075E04 1.350213E02
5 1.796037E04 1.340163E02
6 1.794774E04 1.339692E02
7 1.794722E04 1.339672E02
8 1.794721E04 1.339672E02
9 1.794722E04 1.339673E02
10 1.794719E04 1.339671E02
84
EXHAUSTIVE SEARCH
I OF DATA POINTS = 40
I OF ITERATIONS = 10
t OF PARAMETERS = 6
SIGNAL TO NOISE RATIO = 10*5
ITER
6(0) 6(1) 6(2) 6(3) 6(4) 6(51
.5 1 .5444685 2 .4453036 3 .4814392 4 .4967953 5 .4996233 L 1 .8 .2 .4 .9
1.063756 .8797762 3.155838E02 .1966778 .8292298
1.048258 .7728978 .1354619 .3567541 .8815441
1.0041 .7891053 .1830333 .3889794 .8977166
1.000206 .7968065 .1952727 .397592 .9005974
.9999848 .7991458 .19B3499 .3993473 .9010324
0 .5001632 7 .5002416 Q 1.00011 .7997753 .1989958 .3996641 .9010668
1.000177 .79991B6 .1991137 .3997072 .901059
0 .5002471 Q 1.000198 .7999472 .1991304 .3997092 .9010538
7 .5002453 10 1.000204 .799952 .1991313 .3997079 .9010521
.5002443 1.000205 .7999525 .1991309 .3997072 .9010516
ITER SUN OF RESIDUALS R RNS OF R STEPSIZE = .1
l 6.946154E03 8.334359E02
2 7.797317E04 2.79236BE02
3 5.916066E05 7.691597E03
4 4.56192E06 2.135B65E03
5 1.922627E06 1.3B6588E03
6 1.799648E06 1.34151E03
7 1.794914E06 1.339744E03
8 1.794741E06 1.339679E03
9 1.794739E06 1.339679E03
10 1.794714E06 1.339669E03
APPENDIX B
PROGRAM LISTINGS
All the programs in this appendix are written in GWBASIC
language.
* program "Singular value decomposition" is from [15] it
was written in FORTRAN and translated into GWBASIC.
* program "Recursive least squares is from [16], it was also
written in FORTRAN and translated into GWBASIC.
* the major and main part of program "Exhaustive search
algorithm" : subroutine parameters search" was developed by
professor Douglas A. Ross [11].
86
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
iitttimttimtiitttiiiimtumiiitmttittitmuttiitttMitittii
liuttmuittltl SIN6ULAR VALUE DECOMPOSITION mitlltillttlltUItt
iiititttitttitiiiiiitmmmitmmttiittitttmtitmmtttttitit
'DEFDBL Q,8,E,R)T,Y,U,0,P,A,S)D,F,9,H,Z,1,C1V)B
H=30 : L=20: MHH=ML
Old X(l1)l0(lt)l6(L)lY(Lll)
FOR X=0 TO H1
X(X)=2l(RND.5)ISQR(3J.l)
NEXT X
8(0 )=.5 : 8(1 )=1 :6(2 ) = .8 :B(3 )=.2
6(4 )=.4 : 6(5 )=.9 :6(6 ) =.B :6(7 )=.4
6(8 )=.3 : 6(9 )=l :6(10) =.75 :6(11)=1.5
6(121=1.1 : 6(13)=.6 :6(14) =1.2 :6(15)=.12
6(161=1.55 : 6(17)=1.21 :6(1B) =.66 :6(19)=1.9
FOR X=Ll TO H1
D(X)= 6(0)IX(X)+6(I)II(Xl)+6(2)IXX2)+6(3)IX(X3)+6(4)IX(X4)
D(K)= D(K)+6(5)IX(X5)+6(6)tXX6)+6(7)IXX7)+6(B)IXX8)
D(X] = D(X)+6(9)IXlX9)+6(10)tX(X10)+6(ll)IX(Xll)+6(12)IX(X12)
D(K1= D(X)+6(13)IX(X13)+6(14)II(X14)+6(15)IX(X15)+616)tX(X16)
D(X)= D(X)+6(17)IX(X17)+6(18)IX(X18)+6(19)U(X19)
NEXT X
ttmttmtttttuuit compute r & p uiiMimiiitutttmiitimm
DIM P(Lltl)fR(Ll,Ll)lA(LlL)lAS(LvL)
XX=H1
FOR 1=0 TO Ll
T=0
FOR J=L1 TO XX
T=T+D(J)IX(JI)
NEXT J
P(I,1)=T : Y(I+111)=P(I,1)
NEXT I
FOR 1=0 TO Ll
FOR J=I TO Ll
T=0
FOR J1=(L1)I TO (XXI)
T=T+X(J1)X(JlJ+1)
NEXT J1
H(I,J)=T : R(J,I)=R(I,J)
A(I+1,J+1)=R(I,J) : A(J+1,1+1)= A(1+1,J+l)
NEXT J
NEXT 1
tiumimimtu print p and r itmiuimtmiiiiitiimttiiiii
PRINT 1 THE ELEMENTS OF (Ll)ll MATRIX P ARE (L ELEMENTS):
PRINT THE ELEMENT OF MATRIX P STARTS (0,1) *
FOR 1=0 TO Ll
PRINT P(I,1)
PRINT
NEXT I
PRINT THE ELEMENTS OF MATRIX (Ll)l(Ll) R ARE (LIL ELEMENTS):
87
500 PRINT 1 THE ELEMENT OF MATRIX R STARTS (0,0)
510 FOR 1=0 TO Ll
520 FOR J=0 TO Ll
530 PRINT R(I,J),
540 NEXT J
550 PRINT
560 NEXT I
570 ENTER A.d.N.NU.NV.S.U.V.IP
580 Ittlllltltlltim SINGULAR VALUE DECOMPOSITION IllltltUttltlltlUltl
590 itiitmimuitituim a=uisiv(H) uitmttiiiumitimiimiuti
600 'lltllttllltmi A=HtN MATRIX U=MIM S=NIN V= NtN lltltttltlttttl
610 H=L : N=L : NU=n : NV=N : IP=0
620 HHAX=H : NHAX=N
630 Old U(M,f1),V(N,N),UH(n,H),VH(N,N),SM(M,N),I5n(N,M)
640 DIN S(N)tB(100),C[100),T(100)
650 ETA= 1.2E07 : TOL= 2.4E32
660 NP=N+IP :Nl=N+I
670 FOR 1=1 TO H
680 FOR J=1 TO N
690 AS(I,J)= A(i,J)
700 NEXT J
710 NEXT I
720 ittiimuttmttiiitii household reduction ummiimmitumti
730 C(1)=0
740 K=1
750 Kl=X*l
760 'tmitimimitt elimination of A(I,k),i=k+i, ... ,n nmniimm
770 Z=0
7B0 FOR I=K TO N
790 Z=Z*A(I,K)A2
800 NEXT I
BIO B(X)=0
820 IF Z<=TOL GOTO 1060
830 Z=SQR(Z)
840 B(K)=Z
B50 H=ABS(A(K,K))
860 Q=1
870 IF WHO THEN 0=A(K,K)/H
880 A(K,K)=QI(Z+H)
890 IF K=NP 60T0 1060
900 FOR J=ri TO NP
910 Q=0
920 FOR I=K TO H
930. Q=Q+A(I,K)IA(1,J)
940 NEXT I
950 0=Q/(ZI(Z+N))
960 FOR I=K TO M
970 A(I,J)=A(I,J)QtA(I,K)
980 NEXT I
88
990 NEIT J
1000 'tmitiittittitmttt phase transform!ion tmiuttiMtitmitimi
1010 Q=A(K,K)/ABS(A(KfK))
1020 FOR JKl TO NP
1030 A(K,J)=QIA(K,J)
1040 NEXT J
1050 imtttmtmttt elimination df a(k.jj,j=k+2, ... ,n ttimtttiiut
1060 IF KN SOTO 1370
1070 Z=Q
1000 FOR J=K1 TO N
1090
1100 NEXT J
1110 C(K1)=0
1120 IF Z<= TOL SOTO 1340
1130 Z=SQR(Z)
1140 C(K1)=Z
1150 H=ABS(A(K,K1))
1160 0=1
1170 IF NXO THEN 0=A(K,K1)/H
1100 A(K,K1)=QI(Z+N)
1190 FOR Ml TO H
1200 0=0
1210 FOR J=K1 TO N
1220 Q=Q+A(K,J)IA(I,J)
1230 NEIT J
1240 Q=Q/(ZI(Z+N))
1250 FOR J=K1 TO N
1260 A(I,J)=A(I,J)OIA(K,J)
1270 NEXT J
1200 NEXT I
1290 tltltmuilltmilliuit PHASE TRANSFORMATION tt*tttItItlt11Itllttll
1300 Q=A(K,K1)/ABS(A(K,X1))
1310 FOR I=K1 TO n
1320 A(I,K1)=A(I,K1)IQ
1330 NEXT I
1340 K=K1
1350 60T0 750
1360 tlSIttlttttUtti TOLERENCE FOR NEGLIGIBLE ELEMENTS tlttlltlltltltltill
1370 EPS=0
1300 FOR K=1 TO N
1390 S(K)=B(K)
1400 T(KKIK)
1410 IF (S(K)+TK)) >EPS THEN EPS=S(K)+T(K)
1420 NEXT X
1430 EPS=EPSIETA
1440 'tUlllimtltlllllll INITIALIZATION OF U AND V tmUtttUlltlttSItlltt
1450 IF NU=0 GOTO 1520
1460 FOR J=1 TO NU
1470 FOR 1=1 TO M
89
1400 U(l,J):0
1490 NEXT I
1500 U(J,J)=1
1510 NEXT J
1520' IF NV=0 GOTO 1600
1530 FOR J1 TO NV
1540 FOR 1=1 TO N
1550 V(I,J)=0
1560 NEXT I
1570 V(J,J)=1
1580 NEXT J
1590 'linitUltmtltlllll OR DIA60NALI2ATI0N llllttlttlltlltltmtltllllt
1600 FOR Klt=l TO N
1610 K=N1KK
1620 'tlliutliutmtllllll TEST FOR SPILT MIUIUMItlMIttlltUttlUII
1630 FOR LL=1 TO K
1640 L=K+1LL
1650 IF ABS(T(L))<=EPS THEN 60T0 1970
1660 IF ABS(S(L1))<=EPS GOTO 1690
1670 NEXT LL
1680 tiMititimtiiimmt cancelation of bid mitimmitttmtmu
1690 CS=0
1700 SN=1
1710 L1=L1
1720 FOR I=L TO X
1730 F=SNfT(I)
1740 T(IJCSIT(I)
1750 IF ABS(F)<=EPS GOTO 1970
1760 H=S(11
1770 N=SQR(FA2+HA2)
1780 S(I)SN
1790 CS=H/N
1800 SN=F/N
1810 IF NU=0 60T0 1880
1820 FOR J=1 TO N
1830 X=U(J,L1)
1B40 Y=U(J,I)
1850 U(J,L1)=XICS+Y4SN
1860 U(J,()sYICSXtSN
1870 NEXT J
1880 IF NP=N GOTO 1950
1890 FOR J=N1 TO NP
1900 0=A(L1,J)
1910 R=A(I,J)
1920 A(L1,J)=0ICS+RISN
1930 A(I,J)=RICSOISN
1940 NEXT J
1950 NEXT I
1960 tlltKIIIt TEST FOR CONVERGENCE IllUtllUUlimtllltllllllltllimil

PAGE 1
ADAPTIVE EQUALIZATION byNamHo B.S., University of Colorado, 1986 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of the Master of Science Department of Electrical Engineering 1989
PAGE 2
To the_ GRADUATE SCHOOL OF THE UNIVERSITY OF COLORADO: Student's name: ______ Degree : ..... M==s ....... Discipline : Electrical Engineering Thesis Title : _____ ___ The final copy of this thesis has been examined by the undersigned, and .we fmd that both the content and the form meet acceptable presentation standards of scholarly work in the above discipline. Chairperson dol/ I ftfZ Date
PAGE 3
This thesis for the Master of Science degree by NamDaiHo has been approved for the Department of Electrical Engineering by Douglas A. Ross '1 i'lliam J. Wolfe
PAGE 4
Ho, Nam D. (M.S., Electrical Engineering). Adaptive Equalization. Thesis directe4 by Associate Professor Douglas A. Ross. In adaptive signal processing, one usually encounters identification or filtering problems. Consider the performance function J: J = 1: [e(i)]2 = 1: [d(i)y(i)]2 where e(i) is the difference between the output d(i) of the system (or the plant) and the output y(i) of the model. If the model is a MA model then J has one and only one minimum point and its shape looks like a parabola in one dimension and a bowl in N dimensions (N >2). The objective of an identification problem (to find the parameter values of the model so that they are as close to those of the system as possible) will be achieved simultaneously with the objective of a filtering problem (to minimize J). If the shape of J is flat at the mtmmum point then Exhaustive Search technique (ES) is probably the best way to cope with this situation. ES is easy to understand, it offers very fast
PAGE 5
v convergence and gives very accurate estimates. This thesis will discuss several techniques to solve identification and filtering problems and ES will be investigated in detail. The form and content of this abstract are approved. I recommend its publication. Signed Douglas A. Ross
PAGE 6
ACKNOWLEDGEMENTS P would like to express my smcere appreciation to Dr. Douglas A. Ross for his support and encouragement during the preparation of this thesis. Special thanks are gtven to the student's parents, brothers, sisters and friends for their support.
PAGE 7
TABLE OF CONTENTS CHAPTER I. IN'TRODUCTION ................................................... 1 II. GENERALIZED INVERSE ...................................... 4 21 AR, MA and ARMA models . . . . . . . . . . 4 22 Formulation of Normal equation ................. 7 23 Solving Normal equation ........................... 10 23.1 Direct Inversion ...................... 10 23.2 Gauss elimination ...................... 11 23.3 Iterative technique .................... 11 23.4 Generalized Inverse .................. 12 24 Computer simulation ................... ............ 15 25 Conclusion .................................. .... ... .... 17 ill. RECURSIVE LEAST SQUARES ALGORITHM.... 18 31 Introduction ........................................... 18 32 Formulation of the problem ...................... 18 33 Recursive Least Squares algorithm ............ 20 34 Some properties of RLS ........................... 21 35 Computer simulation .. .... ... .... .... .... ..... ... 23 36 Conclusion ........................... .................... 31 IV. EXHAUSTIVE SEARCH TECHNIQUE (ES) .......... 32 41 Introduction .... .................... .... ...... .... ..... 32 42 Exhaustive Search technique ................... 34 43 Computation ............................................ 44
PAGE 8
Vll a) Exhau_stive Search ...... ..... ... ..... ... ........ .. 44 b) Recursive Least Squares ... .. . .. .. . .. . . 45 44 Sample size .... ... .. ..... .. .. . .. . .. . .. ... .. ... . 48 45 Step size .. ...................................... .......... ... 49 46 Input and output ... ...... ... .... ... ... ... .... ... ... . . 50 47 Noisy MA process........................................ 51 48 Over/Underestimate the order of the system.... 58 a) Overestimate the order of the system ...... 62 b) Underestimate the order of the system .... 62 49 Exhaustive Search for infinite impulse response (IIR) filter ................................... 64 410 Conclusion . .. .. . .. . . . . .. . . . . . . . . . .. . .. . 66 V. DISCUSSION AND CONCLUSION....................... 68 BffiLIOGRAPHY ....................................................... 70 APPENDIX A. THE PARAMETER VALUES OF THE WEIGHT VECTOR OF THE SYSTEM, THE ESTIMATES OF THE WEIGHT VECTOR OF THE MODEL, SUM OF RESIDUALS R, R.M.S OF R ... OF THE GRAPHS IN CHAPTERS II, III, IV ... .. .. . .. .. . .. .. . .. ... .. .. 72 B. PROGRAM LISTINGS ....................................... 85
PAGE 9
v 111 TABLE OF SYl\ffiOLS RLS = recursive least squares ES exhaustive search D number of data points I number of iterations L = number of parameters SIN = signal to noise ratio = step size R sum of residuals
PAGE 10
TABLE OF FIGURES FIGURE 21.1 22.1 35.1 Stochasic model .. .. .. .. .. .. .. .. .... .. .. .. .. .. .. .. ... .. .. .. .. . .. .. 5 Schematic diagram for identification or filtering problems 7 RLS 0=150 1=150 L=6 S/N=infinite parameter values vs number of iterations .. ............ 25 35.2 RLS 0=1000 1=1000 L=6 S/N=infinite R & log10R vs number of iterations ......... ..... ..... .. 26 35.3 RLS 0=1000 1=1000 L=6 S/N=10 parameter values vs number of iterations ..... ... .. .. .. 27 35.4 RLS 0=1000 1=1000 L=6 parameter values vs number of iterations ... .. .. .. .. .. 28 35.5 RLS 0=1000 1=1000 L=6 S/N=10S parameter values vs number of iterations .......... .. .. 29 35.6 RLS 0=1000 1=1000 L=6 S/N=IO, IQ3, 105 R & log10R vs number of iterations ..................... 41.1 The "parabola" shape of the performance function J in one d . un.enston .................................................................... 42 Schematic diagram of exhaustive search ......................... 42.1 ES 0 =50 1=10 L=6 S/N=infinite 30 33 35 dS = .1 parameter values vs number of iterations 40 42.2 ES 0 =50 1=10 L=6 S/N=infinite dS = .I R & log10R vs number of iterations....... 41
PAGE 11
X 42.3 ES D =50 1=10 L=11 S/N=infinite = .1 parameter values vs number of iterations 42 42.4 ES D =50 1=10 L=11 S/N=infinite = .1 R & log10R vs number of iterations ........ 43 47.1 ES D =50 1=10 L=6 S/N=10 = .1 parameter values vs number of iterations 54 47.2 ES D =50 1=10 L=6 S/N=103 = .1 parameter values vs number of iterations 55 47.3 ES D =50 1=10 L=6 SIN=10S = .1 parameter values vs number of iterations 56 47.4 ES D=50 1=10 L=6 S/N=10, 103, 105 R & log10R vs number of iterations ...................... 57
PAGE 12
CHAPTER I INTRODUCTION The field of recursive identification is often viewed as a long and confusing list of methods and tricks. Methods and algorithms have been developed in different areas with different applications. The term "recursive identification" is taken from the control literature. In statistical literature the field is usually called "sequential parameter estimation," and in signal processing the methods are known as "adaptive algorithms." Systems consist of a wide range of objects whose behavior we are interested in studying, affecting or controlling. The typical tasks are related to the study and use of the systems: control, signal processing (filter designing) and prediction. Many techniques have been developed in control theory, communications, signal processing and statistics for solving the problems involving these tasks. In order to use these techniques, some knowledge of the properties of a system i.e., model, must be known. The model of a system can be represented in one of several different forms. For examples, graphics models ( properties of the systems summarized in graphics or in tables), mathematical models ( mathematical relationships between certain variables of a system) are very important and necessary when complex design problems are involved.
PAGE 13
2 In this thesis, mathematical models: moving average (MA) and autoregressive moving average ( ARMA ) will be used. Basically, there are two approaches to building a mathematical model of a given system: the first way is to look at physical laws and relationships between variables inside the system; from there the model is constructed. Unfortunately, this is not always possible: it is timeconsuming and may lead to an unnecessarily complex model. In addition, the system's properties may change in an unpredictable manner. The second way is to use the signal produced by the system; it consists of two general steps: choose a particular member of a family of candidate models, then use an algorithm to estimate the parameters of this candidate. The fundamental theorem in the decomposition of stationary time series proposed by Wold [1] says: The most general decomposition of a stochastic process is a moving average one (MA). An MA process can also be modeled by AR or ARMA; a finite order MA process requires an infinite order AR representation. AR and MA are special cases of ARMA. The correct choice of AR, MA, ARMA and their orders is very important. Mismodeling may lead to serious errors. Akaike information criterion (AIC) and Final prediction error (FPE) are among the choices for estimating the orders. Estimating (identifying) the parameters is usually done in two ways: a batch of data is collected first and used to identify the parameters; this is called offline or batch processing identification. Online or real time identification means updating the estimates of parameters as new data are available.
PAGE 14
3 The following 1s the synopsis of the thesis's content and order: chapter II Generalized Inverse This is an offline technique. The identification problem which leads to normal equation is formulated. Several methods such as direct inversion, Gausselimination, and iterative technique are briefly discussed and finally generalized inverse is investigated. chapter ill Recursive Least Squares (RLS) Here is an online algorithm used for MA orily. RLS algorithm has been investigated extensively in technical literature and is mentioned here for the purpose of pointing out its pros and cons and showing, later on, how exhaustive search solves some of RLS's problems. chapter IV Exhaustive Search (ES) One of the problems associated with RLS occurs when the autocorrelation matrix R in normal equation RW=P is illconditioned. This is why ES came into being. ES works quite well for finite impulse response (FIR) filters but not so for infinite impulse response (ITR) filters. Number of computations, rate of convergence, etc., of ES are investigated in detail in this chapter. chapter V Discussion and Conclusion
PAGE 15
CHAPTER IT GENERALIZED INVERSE 21 AR. MA and ARMA models Many discretetime random processes encountered in practice are well approximated by a rational transfer function model. The idea of representation of a discretetime random process by a model is that a random process consisting of highly correlated observations may be generated by applying a series of statistically independent shocks," white noise, for instance, to a model (discretetime linear filter. figure 21.1 on next page). H an input driving sequence x(i) and. the output sequence y(i) that is to rriodel the discretetime random process are related by the linear difference equation: y(i) Where q L b(k)x(ik) k=O p L a(k)y(ik) k=l a(O), a(l), a(2), ... a(p ), b(O), b(l), ... b(q) are constants, this general model is termed an autoregressive moving average (ARMA) model. The transfer function H(z) between input x(i) and output y(i) is a rational function: H(z) B(z)/A(z)
PAGE 16
White noise Input sequence x(i) ... MODEL Discretetime linear filter with transfer function H(z) figure 21.1 : Stochastic model ... Discretetime random process Output sequence y(i) 5
PAGE 17
6 p q A(z) = L a(k)zk B(z) = L b(k)zk k=O k=O It is assumed that A(z) has all its zeros within the unit circle of the z plane so that the filter is stable. If all the a(k) coefficients except a(O) = 1 vanish for ARMA parameters then: y(i) = q L b(k)x(ik) k=() the discretetime random process y(i) is a moving average of .order q; it is denoted as MA(q) process. The model is called MA or allzero model. Likewise, if all the b(k) coefficients except b(O) are zero in ARMA model then: p y(i) L a(k)y(ik) + b(O)x(i) k=l The output sequence y(i) is now an autoregressive process of order p; it is denoted as AR(p) process and the model is an AR or all pole model. Note that MA model is often known as finite impulse response (FIR) filter while AR, ARMA models are usually known as infinite impulse response (llR) filters [2], [3].
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22 of Normal eguation Consider the following figure: x(i) SYSTEM d(i) e(i) = d(i)y(i) MODEL y(i) figure 22.1 : Schematic diagram for identification or filtering problems. Model : one can use AR, MA, ARMA, etc. In this case MA model (or FIR filter) is used. x(i) input d(i) : desired output or output of the system. y(i) : filtered output or output of the model. Assume the system itself is a FIR filter. The impulse response vectors of the system and the model are: 7
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W.* L respectively. 8 In identification problems, the objective is to obtain W L so that .W L is as close to W L as possible. In filtering problems, minimizing the sum of squared errors is the goal. The objective now is to find the "best" W L such that the sum of squared difference between d(i) and y(i): k k J L [e(i)]2 1: [d(i)y(i)l2 i=L1 i=L1 is as small as possible. J is called the performance function (criterion function). For a given sequence of input vectors { X(i)} and scalars { d(i)}, J is a function of W L only and therefore is a measure of how well W L performs to produce y(i) which matches d(i). The choice of W L minimizing J is the value that has the best performance. Let the input vector be: XL(i) [ x(i) x(i1) ... x(iL+1) ]T and the given data be: x(O), x(1), ... x(k), d(L1), d(L), ... d(k), then
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J = J note: k k L [e(i)]2 = l: [d(i)y(i)] 2 i=L1 i=L1 k k L [d(i)]2 WLT L XL(i)d(i) i=L1 i=L1 k + WLT[l:XL(i) XLT(i) ]WL i=L1 y(i) = 9 k [L d(i)XL T(i)]W L i=L1 Define the autocorrelation matrix R and the correlation matrix P as following: k R = p = L d(i)XL(i) i=L1 i=L1 expand R and P out and write in more explicit form: k k :E x(i)x(i) :E x(i)x(i1) i=L1 i=L1 k1 R :E x(i)x(i) i=L2 k :E x(i)x(iL+1) i=L1 k1 :E x(i)x(iL+2) i=L2 kL+l :E x(i)x(i) i=O
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10 k :E d(i)x(i) i=Ll k p = :E d(i)x(i1) i=Ll k J :I: [d(i)J2 wTp pTw + wTRw i=Ll k J L [d(i)]2 2WTP + wTRW Where w = WL i=Ll To minimize J, take the derivative of J with respect toW and set it equal to zero: dJ/dW = 2P + 2RW = 0 RW = P (22.1) is called the normal equation. 23 Solving the normal equation 23.1 Direct inversion (22.1) The most straightforward approach to solving 22.1 IS the.
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11 direct inversion of R followed by its multiplication with P. The inverse of R is defmed as: R1 adjoint R/det R While this formulation is fine for theoretical work, it is hardly adequate for numerical work. As the order of R increases, the number of computations of R1 gets out of hand; the limitations on word length of the machine used and the numerical accuracy of computational methods also introduce roundoff errors which become significant if the number of computations is large. 23.2 Gausselimination Gausselimination (or sucessive elimination) consists of reducing the system of L equations in L unknowns (for example ) to a system of (L 1) equations in (L1) unknowns by using one of the equations to eliminate one of the unknowns. This process repeats until only one equation and one unknown are obtained. Once this last unknown is found, the remaining (L1) unknowns are calculated by back substitution. If matrix R does not have full rank, R 1 does not exist, direct inversion fails and so does Gausselimination. 23.3 Iterative approximation Given autocorrelation matrix R and correlation matrix P, the estimate of WL at (k+1) th iteration may be computed according to this scheme: WL(k+1) WL(k+1) = [ w 0(k+l) W1 (k+l) ... wL_1 (k+l) JT = [ 1J.!R]. w L(k) + JlP w L(O) = 0 Jl : small positive number
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I: identity matrix It has been shown in [ 4] that if J.l is chosen so that: lim [ lJ.LR]k? 0 k> infinity then lim WL(k) >WL* k> infinity 12 This scheme is not the only way. Others are steepestdescent, Newton methods and their modifications. 23.4 Generalized Inverse Consider again equation (22.1): RW = P R is a linear transformation with domain and range spaces in matrix theory. By introducing arbitrary orthornormal bases for both domain and range, a simple representation (i.e., diagonal matrix) can be found for R. Let: v u = = ( Yt V2 ... VL] ( llt U2 ... UL] be new orthornormal sets of vectors used as bases for both domain and range respectively. The new V: W' . w In range space the new P' of P : coordinates W' of W with respect to VW' P' = UHP :. P = UP' P'
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In the new system, R becomes R' with: R' UHRV :o R UR'VH where R' = :] diag[cs1 CJ2 . cs") k by k matrix I yHy I = V(R'HR')VH 0 RHRv. = 0 0 1 U(R'R'H) UH 0 RRHu. 0 0 I 13 cr.2v. 1 1 cri2 : eigenvalues of RHR and RRHo vi and ui are eigenvectors of RHR andRRH The positive square root of cr? i = 1, 2, 3 ... k, are singular values ofR. U and V are not unique because they consist of ui vi not uniquely determined. To solve RW =Pin the least squares sense means to find W so that IIRWPII2 is minimum. II RWPII 2 = II UR'VHWP 112 = II UR'VHWUUHP 112 Let Y = yHw and P' = uHp The problem now is to minimize IIR'Y P'll 2
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P' = [pl' P2' ... PL']T II R'YP'II2 = [( crlylpl') 2+ ( cr2y2p2')2 + ( O'iYCPi')2 ]112 i = 1, 2 ... L Consider 2 cases: Case I If R is nonsingular then cri is not equal to zerofor i = 1, 2, 3 ... L. By letting cr.y.p. = 0 :. y. = p.'/ cr. I I I I I I and W = VY Case II 14 IfR is singular, at least one of cri must be zero. For instance cri = o for i = k+1, k+2 ... L, then II R'YP'II2 = [( crlylp1') 2+ ( cr2y2p2')2 ... + ( O'iYCPi') + P + p ] 1/2 k+1 ... L let Yi = pi'/ O'i for i = 1, 2, ... k and yi = 0 for i > k :. W = VY Whether R is singular or not, the solution W always exists. If one lets R = diag [ cr 1 1 cr2 1 .. cr x: I] 1 < k < L W = VY = VR*P' = VR*UHP = R+P R+ = VR *UH is called the generalized inverse of R.
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15 There are several algorithms to calculate R +, and they fall 1nto two broad groups. The first group consists of full rank factorization and singular value decomposition (SVD); the iterative techniqQes are. in the second group. One of the commonly discussed iterative techniques in the second group is due to BenIsrael [5], this and its variants are not competitive for large matrices with SVD. The reader interested in the first group is referred to [ 6] for full rank factorization and to [7] for details on SVD. This group could be divided into two smaller ones. Full rank factorization and other methods based on some variations of Gausselimination, call these the "elimination methods" group. The second one consists of SVD only. If one suspects R is severly illconditioned and /or very high accuracy is needed, the best idea of all is to compute in exact arithmetic. This will provide R exactly if R has rational entries. Elimination methods are preferable with exact methods. 24 Computer simulation The program used in this chapter is written 1n BASIC language. A major part of the program: the subroutine Complex singular value decomposition was developed by P .A Businger and G.H. Golub (Communications of ACM Vol. 12, pp. 564565, October 1696) in FORTRAN and was translated into BASIC. The input x(i) is generated as following: x(i) = 2*(RND.5)*SQR(3*.1) where is multiplication, SQR() = ( )1 2 and RND is a function which generates random numbers between 0 and 1.
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16 x(i) is white noise with mean= 0 and variance= .1. The desired output d(i) is generated by applying x(i) to a MA model. The relationship between d(i) and x(i) is : d(i) = g(O)x(i) + g(1)x(i1) + ... + g(p)x(ip). Where g(O) = .5, g(1) = 1, g(2) = .8 ... See the program Singular Value Decomposition in appendix B for values of g(i), i = 0, 1, 2, ... 20. These are the actual values of the weight vector W L of the system. The results in appendix A are the actual values of g(O), g(1), ... g(p) and their estimates for the cases : p = 6, 11, 20. Given input sequence x(i) and output sequence d(i), the program computes the autocorrelation matrix R and the correlation matrix P in the normal equation R W = P, then the generalized inverse of R (R+) is calculated, and finally the solution W IS: W=R+p Several observations should be pointed out: The estimates are fairly accurate even when the number of data points used is small. As the number of parameters (p) increases, so do the number of data to ensure accuracy. The values of singular values a i of R are machine dependent. Numeric values can be either integer, singleprecision or doubleprecision. Singleprecision numeric values are stored with 7 digits (although only 6 may be accurate) and doubleprecision numeric values are stored with 17 digits of precision and printed in as many as 16 digits.
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17 Therefore, if one uses singleprecision and one of the singular values of matrix R happens to be less than 1 o8 then concluding R is singular is wrong. This could happen if doubleprecision is used and one of the singular values is less than 1 o17 Assume that noise e(i) = 2*(RND.5)*SQR(3*D) is added to the desired output d(i). The signal to noise ratio is defined as: Signal/noise = (variance of signal)/(variance of noise) where variance of signal d(i) = .29 (computed by using 100,000 data points ) and variance of noise e(i) is D By varying D, different signal to noise ratios will be obtained. The estimates are quite good when signal to noise ratio is high as expected. The data when signal/noise = 10, 103 105 are in appendix A. 25 Conclusion This chapter swnmarizes several various techniques to solve least squares problems (RW=P), or equivalently to find the best impulse response vector W of FIR filter so that sum of squared errors are minimized. Direct inversion, Gausselimination, iterative techniques and generalized inverse are briefly discussed. When matrix R in the normal equation RW=P is singular, only iterative and generalized inverse methods are applicable. There are several algorithms to compute generalized inverse of a matrix. SVD is one of them and is widely used. "elimination methods" are recommended when a matrix is illconditioned and /or high accuracy is needed.
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CHAPTER ill RECURSIVE LEAST SQUARES (RLS) 31 Introduction There are situations in which as new data are available, the impulse response vector W = W L needs to be updated accordingly (adaptive filters), but this becomes frustrated and timeconsuming if one uses batch processing methods discussed in chapter II since R I (direct inversion) or R+ (generalized inverse) ... has to be recalculated. This is one of the motivations for introducing recursive least squares algorithm (RLS). 32 Formulation of the problem The formulation of the problem for RLS is proceeded exactly in the same way in chapter II. Recall equation (22.1) RW = P redefined: k1 R = L X(i)XT(i) = RK i=L1 k1 p = 1: X(i)d(i) i=L1 Assume new data x(k) and d(k) now are available. How should one update W recursively ?
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19 k RK+1 = L X(i)XT(i) = RK+ X(k)XT(k) i=L1 k PK+1 l: X(i)d(i) P K + X(k)d(k) i=L1 WK+1 RK+11 PK+1 Employing the well known matrix inversion lemma, sometimes called "ABCD lemma": ( A + B C D t1 = A1 A1 B [ D A 1 B + c1 r1 D A1 This identity is proved by multiplying the left with the right side and getting identity matrix I Applying "ABCD lemma" for RK+1 with the following associations: Then R 1 K+1 A = RK B = X(k) = R 1 K C 1 D = XT (k)
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20 Note R lp. K K y(k) Let qk (scalar) e(k) d(k)y(k) v After some algebraic manipulation and simplification: 33 Recursive least sguares algorithm CRLS): RLS consists of the following steps [8] : 1) Accept new data x(k), d(k). 2) Form X(k) by shifting x(k) into the information vector. 3) Compute the prior output (estimated output) y(k) = XT(k) WK 4) Compute the error e(k) e(k) = d(k)y(k)
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21 5) Compute the filtered information vector ZK 6) Compute. the scalar qk 7) Compute the gain constant v 8) Compute the normalized filtered information vector ZK = 9) Update the weight vector 10) Update the inverse of autocorrelation matrix R 1 k+l 34 Some properties of RLS = 1) To use RLS, first of all, the autocorrelation matrix Rand the weight vector W must be initialized. For the weight vector, one could choose: W(O) = 0 For the autocorrelation matrix Rk' the initial value R0 can be computed in this way:
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k RK+1 L X(i)XT(i) i=L1 k RK+1 = L X(i)XT(i) i=L2 Ro = oi with I = identity matrix 0 small positive constant R 1 s::11 0 u 22 + oi 2) The introduction of "forgetting factor" p into RLS ( 0 < p < 1) will put more weight on the recent data and less weight on the older ones. The motivation of using p comes from the cases where the data { X,d} changes its charateristic within record of data points (nonstationary process). The RLS in this case will be [9] : R 1 K+1 pk+1 = ] p [ (1/ p) d(k)X(k) + Pk]
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23 d(k)y(k) = vvk + zk p = 0 then every data is weighted equally. 3) From [10] one of the statistical properties of RLS is: b(k) 0 R1vv* k k number of iterations R1 : inverse of R VV *, VV : weight vectors of the system and the model respectively. b(k) = E[VV(k)] VV *=the bias b(k).becomes smaller ask increases. \Vhen k approaches infinity then b(k) approaches zero. o should be small to reduce the bias. 35 Computer simulation From figure 22.1 page 7, let: w [wo"' WL1*] = w1 L the impulse response vector of the system VVL = [wo wl WL1] = the impulse response vector of the model L length of both VV L and VV L
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D R = number of data points or number of iterations sum of residuals The desired output d(i) is generated by applying the input x(i) = 2*(RND.5)*SQR(3*.1) to the system The relationship between d(i) and x(i) is: d(i) = w0*x(i) + w1 x(i1) + ... + wL_1 x(iL+1) where w0 = g(O) = .5, w1 = g(1) = 1, w2 = g(2) = .8 ... 24 See the program "recursive least squares" in appendix B for the values of g(i), i = 0, 1, ... 5 RLS is used in this chapter to find W L so that W L gets as close to WL* as possible. Figure 35.1 (page 25) shows the values of the estimates (w0 w1 wL_1 of the vector W) and the values of the parameters ( w0*, w 1 ... wL_1 of the vector W L ). The dotted lines are the estimates and the solid lines are the parameters. This is the case where L = 6 and D = 150. Figure 35.2 (page 26) is the plot of sum of residuals R vs number of iterations D ( D = 1000). Figures 35.3, 35.4, 35.5 (pages 27, 28, 29) show the estimates and the parameters when noise is added to the desired output.d(i) and the signal to noise ratio SIN= 10, 103 10s. L = 6, D = 1000.
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2 1 1 2 parameter values I \ I t r,. I L..J . I t LJ figure 35.1 : # of data points # of iterations 25 61 ,. 121 151 number of iterations recursive least squares = 150 # of parameters = 6 = 150 : signaJJnoise = infmite
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19I1 19I6 19E11 19I16 19I21 R sum of residuals R 6 11 16 1 21 1 LOC(R) 259 figure 35.2 : # of data points # of iterations 26 599 759 number of iterations recursive least squares = 1000 # of parameters = 6 = 1000 : signaVnoise = infinite R = sum of residuals
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27 parameter values .ss 1.1 81 ,. 1111 r""' number of iterations .J figure 35.3 : # of data points # of iterations recursive least squares = 1000 : # of parameters = 6 = 1000 : signaJJnoise = 10
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28 parameter values 1 1 I. .ss .ss 1.1 1\ ___ ...__.,_._, __________________ _,_ I r figure 35.4 : # of data points # of iterations ........ 699 899 1999 number of iterations recursive least squares = 1000 : # of parameters = 6 = 1000 : signaVnoise = 163
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29 parameter values 1.1 \ .55 .ss 1.1 1\_,_ ........ ______________ ,.,_ _____ 811 Hila number of iterations ,, _____________________________ ___ figure 35.5 : # of data points # of iterations recursive least squares = 1000 : # of parameters = 6 = 1000 : signaVnoise = 1 OS
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lQE1 liE3 liE5 19E7 19E9 R 1 3 5 7 9 LOC 30 sum of residuals R signal/noise = 10 signal/noise = 103 signal/noise = 1 OS 1 258 751 1988 number of iterations figure 35.6 : recursive least squares # of data points = 1000 # of parameters = 6 # of iterations = 1000 signal/noise = 10 103 105 : R = sum of residuals
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31 Figure 35.6 (page 30) is the plot of R vs D ( D = 1000) for SIN= 10 103 10s. The data for these graphs are in A. 36 Conclusion RLS algorithm works quite well when no nmse is involved, gives fairly good estimates but needs many iterations. "Fast" RLS is one of the modifications of RLS to reduce the amount of computation.
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CHAPTERN 41 Introduction J Recall the performance function J in chapter II and ill: k L [e(i)]2 i=L1 k = l: [d(i)y(i)]2 i=L1 J is the function of weight vector W only, its shape looks similar to a "parabola" in one dimensional problem (W = [ w 0]) and a "bowl" in L dimensions (W = [ w0 w1 ... wL_1 ]). Figure 41.1 (next page) is the "parabola" shape of J in one dimension, the minimum point of J corresponds to the best impulse response vector W = W optimal = W Recall the normal equation: RW = P In one dimension, the condition of the matrix R determines the shape of the "parabola" at the minimum point If R has full rank and well/illconditioned, then the "parabola" is sharp/flat at this point. Generalized inverse and Exhaustive search (ES) are probably the most suitable ways to cope with R being illconditioned. ES is quite simple and easy to understand.
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J ( performance furiction ) J (minimum) W* w figure 41.1 : The "parabola" shape of the performance function J vs W in one dimension. W* is the best impulse response vector . 33
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34 Although the purpose of RLS and ES is to find the optimal value of the weight vector W (W optimal ), their approaches are different. RLS bases on the concept of updating vector W (k 1) by just the right amount to generate W(k). The idea behind ES is to search the best impulse response vector W optimal on the performance function J. The following sections will examine exhaustive search technique and its properties. 42 Exhaustive search technigue CES) There are several ways to search W optimal on the performance function J: GaussNewton, gradient (results in least mean squares algorithm), and exhaustive search (ES) [11]. For simplicity, consider one dimensional case (figure 42 on next page). Let W = [ W 1 ] and the optimal weight vector W optimal = W"' = [ W 1 "']. Assume one starts with some initial value of W 1' i.e., W1(0). From W1(0) how should one proceed to the new value W1(1) in such a way that W1(1) gets closer to W1 "'. Let the value of J at W1(0) be R1 and W1(0) = S1 then: R1 = J[W1(0)] = J(S1 ) By increasing S1 by an amount two possibilities can happen: J (S1 ) > J (S1 + or J (S1 ) < J (S1 + a) If J (S1 ) > J (S1 +AS), let : R1 = J (S1 ) and R2 = J (S1 +AS) = J (S2 )
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0 J (performance function) W* 1 J .. mmunum figure 42: Schematic diagram of exhaustive search technique in one dimension 35
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36 This situation corresponds with J [W 1 (0)] being on the left side of the parabola representing the performance function J. Therefore, increasing S 1 will make J decrease, it is the hint giving one which direction to go. Next, increase S2 = S1 + 8S an amount 8S, namely: S3 = S2+8S If J (S3 ) < J (S2 ) then R1 = J (S2 ) and R2 = J (S3 ) and increase S3 i.e., S 4 = S3 + 8S evaluate R3 = J (S 4 ) Repeat this procedure until R1 > R2 and R3 > R2 Fitting a parabola R through these three points R1 and R3 : R = a+ bS + cS2 Three unknowns a, b, c are determined if R1 R2 and R3 are known. Take derivative of R with respect to S and set it equal to 0. dR = dS 2cS + b = 0 .. Smin will equal to the new value of W1(0): b 2c Note that the parabola fitting through R1 R2 R3 and the one representing J are not the same and neither are Smin and W 1 *.
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37 = dS = s2sl = s3s2 s2 = s1 +dS s3 = s 2 + .dS s1 + 2dS R = a + bS + cS2 Rl = a + bS1 + cS 2 1 (Sl' Rl) (1) = a + bS 2 + cS 2 2 (S2' (2) a + bS 3 + cS32 < s3, R3) (3) Use Cramer's rule to find b, c from (1), (2), (3) and Smin = (b/2c): 1 Rl S2 1 1 8,_2 1 b = 1 sl S2 1 1 s2 8,_2 1 s3 S2 3
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b 1 sl 1 s2 1 s3 c = 1 sl 1 s2 1 s3 c = s. = mm 1 1 1 1 1 1 RI S2 1 8S [ R1 R2+ R3 ] sl S? s2 S2 2 s3 S2 3 Repeat the above procedure until : 38
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39 b) If J (S1 ) < J (S1 + let: In this case J [ W 1 (0) ] is on the right side of the parabola representing the performance function J. Increasing S will make J increase, going in other direction should decrease J. Next step is to decrease S1 an amount of S3 = S1 and let R3 = J (S3). From here on, the procedure is proceeded exactly the same as in case a). ES can be easily extended to L dimensional problem where: w = [ Wo w1 ... wL1 l In this case each Wi, i= 0, 1, 2, ... L1 is used to minimize J one at a time (W 0 is first, W 1 is next, ... and finally W L1). After this the process is repeated again and again . until J can not be reduced further or W optimal is achieved. Figure 42.1 (page 40) shows the estimates (dotted lines) of the weight vector of the model and the parameters (solid lines) of the weight vector of the system vs number of iterations I. Where: length of vector L = 6 number of data points D = 50 number of iterations I = 10
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40 parameter values 1 .s I 2 8 I \ \ number of iterations \ .5 \ \ 1 figure 42.1 : exhaustive search # of data points = 50 : # of parameters = 6 #of iterations = 10 : stepsize = .1 signal/noise = infinite
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sum of residuals R liE1 1 1QI 6 6 liE11 11 .. 18116 16 lQI21 21 1 R LOC(R) 2 3 4 5 6 1 number of iterations figure 42.2 : exhaustive search 41 8 # of data points = 50 # of parameters = 6 # of iterations = 1 0 : stepsize = .1 li signaVnoise = infinite : R = sum of residuals
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42 parameter values 1 .5 I 5 7 ,u number of iterations .5 1 figure 42.3 : exhaustive search # of data points = 50 # of parameters = 11 # of iterations = 10 : stepsize = .1 signaVnoise = infinite
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43 sum of residuals R 1911 1 19I6 6 19E11 11 lil16 16 19121 21 1 2 3 4 5 7 8 19 R LOC(R) number of iterations figure 42.4 : exhaustive search # of data points = 50 # of parameters = 11 # of iterations = 10 : stepsize = .1 signaVnoise = infinite : R = sum of residuals
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44 Figure 42.2 (page_ 41): plot of sum of residuals R (or J) vs I. Where L = 6, D =50, I= 10. Figure 42.3 (page 42) and 42.4 (page 43): the estimates, the parameter values, R vs I. L= 11, D =50, I= 10. The data of graphs are in appendix A. One observation that should be pointed out is that the rate of convergence of exhaustive search is very fast. Mter only 5 iterations, the estimates are very good and the sum of residuals R is in the order of l09 r.m.s of R is in the order of l05 The generation of input x(i) and output d(i) was done exactly in the same way mentioned in chapter III. 43 Computation This section attempts to compare RLS and ES computationally. a) ES Let L number of parameters of the weight vector. number of data points used. mult. sub. multiplication subtraction Each y(i) ( filtered output) needs: L(mult.) + (L1) (add.) LE [L(mult.) + (L1) (add.)] Each value of R ( R1 R2 R3 ), where: add. : addition div. : division > each y(i) .> y(i)
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45 4: R (1/ Lr:) L [d(i) y(i)]2 requires: i=l 1) (sub.) + (mult.) + 1) (add.) + 1(div.) + Lr: [L(mult.) + (L1) ( add.) ] or: LE (L+ 1) number of mult. and LE (L+ 1) 1 number of add. and sub. Let A : average number of times when the situation R1 > R2 and R3> R2 occurs. B : number of iterations for ES ceases to work (optimal values have been achieved). The total number of computations for using ES is: 3.(A).(B) [ LE (L+ 1)] number of mult. 3.(A).(B) [ Lr: (L+1) 1] number of add. and sub. b) RLS Going back to page 20 and 21, let:
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46 L number of parameters of the weight vector. number of data points used or number of iterations. step 4 and 7 require one sub., one add. and one div. steps 3, 6, 8, 9 needs 4L mult. and [(L1)+(L1)+L] = 3L2 add. step 5 : L 2 mult. and L(L1) add. step 10: L2 mult. and L2 sub. The total number of computations for one iteration: [3]+[4L+3L2]+[2L2 +L2+ L2 L] or [2L2 +4L]+[L2 + L2 L+3L2+3] [2L2 +4L] (mult) + [2L2 +2L+1] (add., and sub.) The total number of calculations for iterations: LR [2L 2 +4L] number of mult. LR[2L2 +2L+1] number of add., sub., div. As far as number of multiplications is concerned: RLS ES lr;[3.AB.L+3.AB]
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47 Most of the while AB > L. Example: assume RLS andES are used to estimate the parameters of the weight W L = [ W0 W 1* ... W L/l with L = 6. The vjues of the parameters are: W*4 4 . RLS recursive least squares needs 1,000 iterations in order to achieve the following results: I :: : Sum of residu1 R = 2.417282E14 w 2 .8999999 W5 = .8999999 r.m.s of R 1.554761E7 The number of multiplications for using RLS in this case is: LR[2.L.L+4.L] = 1000[(2)(6)(6)+4(6)] = 96,000 mult. ES : : ES needs 7 iterations (B=7) to achieve the above similar results,le average number of times when the situation R1> R2 I
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48 and R3 > R2 occurs is A= 12 (actually, A= 82/7). The number of data points used is = 30, the stepsize = 0.1. The following results are obtained by using ES: Wo .5000012 WI .9999988 w2 = .7999998 w3 .1999998 w4 .3999995 ws .8999994 Sum of residuals R = 2.329294E14 r.m.s of R 1.526203E7 The number of multiplications in this case is: = 30[3(82)(6)+3(82)] = 51,660 mult. *) From this example, ES obviously requires less computation than RLS does. *) If adaptive stepsize is used in ES case, then the number of computations can be reduced .further. 44 Sample size Box and Jenkins [12] had suggested about 50 observations (data points) are an adequate sample size and is also the minimum required number to build a model. A too small sample size results in poor estimates. While larger one gives more accurate estimates at the
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49 cost of more computations. However, computer experiments had shown 50 data points are more than enough, for the cases of L = 6 and 21 (L : number of parameters) even 20 and 30 data points respectively are sufficient. There is no rule of thumb how big the sample size should be, as L increases so does the sample size. 45 Stepsize The speed of convergence and the accuracy depend very much on the stepsize Large makes estimates converge fast to optimal values but the accuracy is sacrified; small has opposite effect. The best approach to compromising these two ( speed of convergence and the accuracy) is to use adaptive stepsize. In the beginning of ES, large is used to ensure fast convergence, as the estimates get closer to the optimal values, small is used (one can choose new equal to onehalf, onetenth ... of the previous one) to achieve desired accuracy. When the matrix R in normal equation RW =Pis suspected to be illconditioned, small6S is highly recommended. Most of the results obtained in this chapter use 6S = 0.1 Again there is no specific rule as to what the proper .6S should be. 6S may be as large as 0.1, 1 or as small as los lots (this depends on machine being used, single/doubleprecision).
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50 46 Input and desired outtmt It is fairly obvious that some conditions on the input sequence x(i) must be introduced in order to secure reasonable identification results. The input must have sufficient rich frequency content to excite all the modes of the system. The exhaustive search technique starts with some initial value ofR: 1 .k R L [e(i)]2 (kL) i=L1 1 k R :L [d(i) y(i)J2 (kL) i=L1 1 k R = :L [d(i)J2 (kL) i=L1 y(i) = 0 i = L1, L, ... k This initial value of R depends on absolute value of mean (lmeanl) and absolute value of variance (lvariancel) of desired output signal d(i) which in some cases can be controlled by using proper input signal x(i) (which is chosen by the user). Small mean and variance of d(i) are desirable, give small initial value of R.
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51 47 Exhaustive search for noisy MA process One of the problems withES which limits its utility is ES's sensitivity to the additive noise ( noise can be added to the input, system or the output). Recall the normal equation: RW = P where k1 k1 R = l: X(i)XT(i) and P = l: X(i)d(i) i=L1 i=L1 When noise is added to the input x(i), the autocorrelation matrix R will change from R to 8R+R. oR is the result of input noise. Likewise, when noise is added to the output y(i), the correlation matrix P will change from P to 8P+P. 8P is the result of both input noise and output noise. Then the solution will change from W to oW+ W. Let c(R) = IIRII IIR111 be the condition number
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52 where II II denotes any of the operator norms of R: II 111 II 112 .. Let R be nonsingular and W be the solution to normal equation RW=P. Then from [13] it had been shown that: 118WII 118PII 118RII < M. c(R). [ + ] (47.1) IIWII II Pll IIRII Although ( 47.1) gives only the upper bound of the ratio 118WII /IIWII it does explain several important points: 1) If 8R and 8P are large then it is reasonable to predict 118WII will be large. 2) Asume that there is no noise added to the input (which is the case for computer simulation in this section). Then 8R = 0, and: and 118WII 118PII < c(R) [ ] upper bound UB IIWII II Pll
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53 UB depends on two terms : c(R) and (118PII I IIPII). *) Assume 118PII I IIPII is a constant. The term c(R) is the condition number of matrix R, small or moderate value of c(R) implies that the equation RW =Pis well conditioned. A large c(R), however, does not imply the equation is illconditioned. (In numerical analysis, if "small" changes in the data lead to "large" changes in the solution, the problem is said to be ill conditioned; otherwise, it is said to be wellconditioned.) Although a large condition number c(R) does not indicate the solution W to equation RW = P is illconditioned for every P, it is true that W is illconditioned for some P. *) Assume c(R) is a constant The term 118PII in 118PII I IIPII corresponds to the additive noise. Large I small noise will lead to large I small 118PII which will make upper bound UB = c(R) ( 118PII I IIPII ) large I small and therefore there will be large I small change 118WII. This explains that as more noise is added to the output (signal to noise ratio decreases ) the estimates become worse. So the change 8W in solution W to normal equation RW = P depends not only on the condition of matrix R (ill/wellconditioned) but also on the "magnitude" of noise (variance of noise). For signal/noise ratio = 10, the estimates are quite different from the actual values of parameters ( figure 47.1 on page 54 5 The estimates are getting better ( the difference between the estimates and the parameters become smaller) as signal/noise ratio increases as indicated by figures 47 .2, 47.3 on pages 55, 56 Figure 47.4 is the plot of sum of residuals R vs number of iterations I for the cases S/N = 10, 103 and 105
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54 parameter values 1 1 I I ,1 ==It I 'l I II .SS l iL, 8 f ..._.,.' n I w a Y:,....4, a a I ., l I number of iterations 1, "\\ .SS \ I ..,. 1.1 figure 47.1 : exhaustive search # of data points = 40 # of parameters = 6 # of iterations = 20 : stepsize = .1 signal/noise = 10
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parameter values 1.1 /. I jl I/ I 55 ..... I{/ ...._. ,, I I 4 55 ala1214nu 28 ""' number of iterations \. \._._, ________ .55 \ \ 1.1 figure 47.2 : # of data points # of iterations exhaustive search = 40 # of parameters = 6 = 20 : stepsize = .1 signaVnoise = l
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parameter values 1.11 I i i. ,I.,_,.1' I' If .ss :j 56 ,1, .... 1,' , ..... 11/ ... '' hi 'j 9 \'.. 4 ' I ., .... number of iterations ', ... \ :. __________________ .55 \ \. I I \
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1iE1 1 liE9 _, R LOC(J) sum of residuals R 1 3 5 1 figure 47.4 : # of data points #of iterations signal/noise = 57 11 13 15 17 21 number of iterations exhaustive search = 40 # of parameters = 6 = 20 : stepsize = .1 10, 103 105 : R = sum of residuals
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58 48 Overestimate/underestimate the order of the system Before getting into the details of the consequences one will get if exhaustive search technique is used for the mode_! having the order which is greater or less than the order of the system, let's look at the modeling technique widely used in statistics, namely BoxJenkins methods. The BoxJenkins modeling procedure consists of three stages for finding a model fitting the available data. stage 1: identification. Choose one of the models among autoregressive (AR), moving average (MA), autoregressive moving average (ARMA) ... models. stage 2: estimation. Estimate the parameters of the model chosen at stage 1. stage 3: diagnostic checking. check the candidate for adequacy. If the model is satisfactory at stage 3, then the procedure stops, otherwise one has to go back to stage 1 or 2 over again until a "good" model is obtained. There are quite a few criterions to make sure a model is a "good" one at stage 3. One criteria is that the model must be able to predict the future value with good accuracy, the other is that a good model must contain the smallest number of parameters. Assume d(n), d(n1 ), ... d(m) are available data and a moving average (MA) model of order L is used to fit the above
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59 data. The relationship between the weight vector W L = [ W 0 W 1 ... W L ] the input white noise x(k) and the filtered output y(k) (output from MA model) is: y(k) W0* x(k) + W1* x(k1) ... + WL* x(kL) The objective of BoxJenkins technique is to find W0*, W/' ... W L so that y(k) and d(k) are as well matched as possible with the smallest value of L. For the details of how to find W 0 W 1 ... W L and the criterions to judge the "goodness" of these values, the interested readers are referred to [14]. One of the criteria is tvalue ( this is an indication of how significantly one estimate is far from zero): The following example will clarify the meaning oftvalue). Example: Take Wa* for instance. For a given data d(n), d(n1), ... d(m) of some process, using BoxJenkins technique, Wa"' is obtained. Given another set of data d((n'), d(n'1), ... d(m') of the same process (one process has many realizations. d(n), d(n1), ... d(m); d((n'), d(n'1), ... d(m') are just two particular ones), another Wa* will be obtained. Wa* is a random variable, it has normal distribution with mean = r (for simplicity, let r = 0) and standard deviation S. At stage 2 (estimation), Wa* is estimated and its tvalue is found (for example) to be 2. What does this This means Wa* is significantly different from r = 0. How is this so?
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60 A rule of thumb in statistics is that only 5% of the possible value W a would fall two or more times standard deviation S away from r = 0. So t = 2 means t = 2 times standard deviation S, and therefore W a is significantly different from r = 0. Finally, this means W a must be included in the MA model to ensure the "goodness" of the model. A model missing W a will lead to bad results such as bad prediction, d(k) and y(k) are not matched, and large sum of residuals. If by mistake, W a is not in the model, stage 3 will verify this and one will go back to stage 2 (or 1) to modify the model. In the BoxJenkins technique, those parameters having small tvalues will have absolute values close to 0, and they do not cons tribute heavily to the value of output y(k) and. hence can be neglected. A parameter is said to be significant in a MA model if it constributes heavily to the value of the output compared with other parameters. A small absolute value of a parameter does not mean that this parameter is not significant. So the term "significant" has relative meaning here. Now let's go back toES. a) Overestimate the order of the system Let L5 and LM be the orders of the system and the model respectively. wLs = [ Wo"' Wt ... wLs J weight vector of the system.
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WI.m [ Wo W1 ... WLm ] weight vector of the model. 61 The desired output d(k) produced by the system and the filtered output y(k) produced by the model are: d(k) y(k) wox(k) +W1x(k1) ... + Wox(k) + wl x(k1) ... + WLm x(kLM) Because all the parameters of W Ls are included in W Lm, by using exhaustive search, one can expect:_ w > w w > w w > w 0 0 1 1 Ls Ls and WLs+1' WLs+2' WI.m >0 and sum of residuals = l:[e(i)] 2 = l:[d(i)y(i)] 2 is minimized. Example: (computer simulation) Let W* = 5 = 5 : the order of the system. [ .5 1 .8 .2 .4 .9] weight vector of the system. 8 : the order of the model. weight vector of the model. The input white noise x(k) and the desired .output d(k) are generated as before in chapter ill. The number of data points = 37. The number of iterations = 10. Using ES, the estimates of the weight
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vector of the model W 8 : [ .5 .9999999 .7999999 6.343689E9 3.395458E8 sum of residuals = 2.34879211 d(k) and y(k) are well matched. b) Underestimate the order of the system 62 .2 .4 .8999999 1.0699998 ] Because not all the coefficients of W Ls are included in W Lm' there will be some coefficients of WLs* left out By using ES, the results of its objectives (to minimize sum of residuals, to have the estimates of the model as close to the parameters of the system as possible, y(k) and d(k) are well matched) depend not on how many parameters are excluded but on how significant these excluding parameters are. The following examples will clarify this: Example1: L5 = 8 : the order of the system W* 8 = [ .5 1 .8 .2 .4 .9 104 2. 104 3.1o5 J weight vector of the system. 5 : the order of the model. weight vector of the model. The input white noise x(k) and the desired output d(k) are generated as before in chapter III. The number of data points = 37. The number of iterations = 10. Using ES, the estimates of the weight vector of the model W 5 is:
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63 [ .4999712 1.000022 .8000126 .1999798 .399953 .9000398 ] sum of residuals R = 3.790102E9 In this example, the last three parameters of the system:104 2. 104 3.105 have very small absolute values, they are much less significant compare4 to other parameters. Therefore, excluding these last three does not lead to bad result at all. In fact, the sum of residuals is quite small (= 3.790102E9), the estimates are close to the parameter values and finally d(k) and y(k) are well matched. Example 2: L5 8 : the order of the system w 8 = [ .5 1 .8 .2 .4 .9 .1 .1 1 ] weight vector of the system. 7 : the order of the model. [Wo wl ... w7 1 weight vector of the model. The estimates of the weight vector of the model W 7 : [ .3050533 1.158459 .8904336 .1458553 .2210489 .8206268 2.065472E2 9.711713E2] sum of residuals R = 7.387126E 2 In this example, W 8 has 8 parameters W 7 has 7 parameters. Although the model misses only one parameter, the results (sum of residuals is large the estimates of the model are not close to the parameters of the system, y(k) and d(k) are not matched) are bad.
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64 This is due to the fact that the excluding parameter ( 1 in this case) is quite significant. Its absolute value (I 11 = 1 ) is more or less equivalent to the absolute values of other parameters). It is noted that the model with a small number of parameters is preferred in practice. However, if this model leads to bad results then one may suspect that some significant coefficients were excluded and therefore increasing the order of the model may be helpful. 49 ES for Infinite impulse response CIIR) filters. Self adjusting or adaptive filters (FIR & IIR) have been successfully applied to a wide spectrum of problems, ranging from adaptive control systems, adaptive estimation, to adaptive signal processing. In a broad sense, most applications in the control and communication areas can be regarded as "signal processing". For the most part, signal processing applications. have relied heavily upon well known and well understood adaptive finite impulse response (FIR) filters. However, in real life, there are common situations where FIR filters are not practical, and lead to heavy computation. As the result, in recent years, research has been directed to extending the adaptive filters to more general, complicated yet more efficient infinite impulse response (IIR) filters. One of the advantages of IIR over FIR is the substantial decrease in computation. Exhaustive search has been shown to work successfully for FIR filter without additive noise or when signal to noise ratio is high. In this section, it is attempted to use ES for IIR filters. ES does not work well for IIR as opposed to FIR filters. Several observations deserve to be pointed out: *) In filtering problems, the objective is to minimize sum
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65 of squared errors L[e(i)]2 = L[d(i)y(i)]2 The estimates of the weight vector of the model and the parameter values of the weight vector of the system may be quite different. These differences may be substantial but are tolerated as long as L[ e(i) ]2 is minimized. In identification problems, the goal is to obtain the estimates as close to the parameter values as possible. For FIR filters, the filtering problems and the dentification problems are achieved simultaneously. This is not so for IIR filters: minimizing sum of squared errors L[e(i)]2 does not make the estimates get close to the parameter values but the inverse is true. *) For MA model (FIR filters), the performance function J = L[e(i)]2 has parabola shape and has one and only one minimum point. On the other hand, for ARMA model (IIR filters), the performance function J has several minimum points. *) To use ES, one must initialize the estimates. Different sets of initial values can lead to different minimum points and in some cases may lead to neither minimizing L[e(i)]2 nor good estimates. Example: The input white noise x(k) = 2*(RND.5)*SQR(3*1) and the desired output d(k) produced by the system are related as following: d(k) = .2x(k1) + .4x(k2) .5x(k3) .9d(k1) + .6d(k2) Likewise, the relationship between x(k) and the output y(k) produced by the model is:
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66 y(k) = W1x(k1) + W2x(k2) + W3x(k3) + W4y(k1) + W5y(k2). Using ES with stepsize = .01, the number data points used= 50 and the initial values of the estimates are chosen to be: wl .18 w2 = .42 w3 .51 w4 = .89 ws = .62 After 40 iterations the following results are obtained: wl .2115997 w2 = .416828 w3 = .5088732 w4 .894435 ws = .5998373 sum of squared errors L[e(i)]2 = L[d(i)y(i)]2 = 1.21141E4 From this example, one can see that although the initial values are intentionally chosen to be close to the parameter values, after 40 iterations the estimates do not converge to the parameter values at all. The sum of the squared errors is not very small. Many iterations are needed 410 Conclusion For MA model (FIR filter), exhaustive search (ES) works quite well when there is no additive noise. In terms of computation and accuracy, ES is better than recursive least squares (RLS). Especially, when the autocorrelation matrix R in normal equation RW =Pis illconditined, ES gives more accurate estimates than RLS does. This is the main reason why ES came into being.
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67 When there is noise involved, none of RLS, ES, generalized inverse works better than the other two. For ARMA model, ES does not work well as explained in section 49.
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CHAPTER V DISCUSSION AND CONCLUSION The frrst step in using one of the techniques in chapters 2, 3, 4 is to assume a model. There are several kinds of models, but only MA and ARMA were used in this thesis. The second step is to estimate the order of the model using criterions such as FPE, AIC, CAT ... In identification problems, the objective is to obtain the estimates as accurately as possible. In filtering problems, one is concerned about minimizing the difference between the desired output ( produced by the system or the plant) and the model's output and can tolerate substantial parameter error. Techniques are classified into two broad groups: offline and online. SVD andES belong to the former. RLS is in the latter group. There is great flexibility in selecting computational methods without any restriction on time, as a result the accuracy of the estimates is fairly high for offline and preferred by statisticians. For engineers, both accuracy and time are crucially important and online techniques are preferred although they have less accuracy. In addition, in many situations one could not afford to waste time to collect all the data. In chapter II, several techniques : direct . Inversion Gausselimination iterative technique, singular value decomposition
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69 were briefly mentioned tp solve normal equation RW = P. When R is singular, all others fail except SVD and iterative techniques. Chapter III was about recursive least squares algorithm (RLS). RLS is based on the concept of starting with some initial values of the weight vector of the model and then using each input sample (input & desired output) to update the estimates so that the estimates get closer and finally converge to the parameter values of the weight vector of the system. RLS works pretty well when there is no additive noise, the estimates are quite good but requires many iterations (which means heavy computation). Chapter IV is about exhaustive search (ES). ES requires less computation and is more accurate than its counterpart RLS. Most of the properties of ES in chapter IV were not proved mathematically but obtained through computer simulation. Computer simulation is a very important and useful tool for investigating ES in particular and any other algorithms in signal processing. Howerver, a serious limitation is that it may not be conclusive. It is difficult to tell whether a simulation result has universal implication or merely reflects of the chosen data sequence. To obtain results of more general validity one must use mathematical analysis. When there is noise involved, none of SVD, RLS ES works better than the other two. With noise free data ES is the best in term of accuracy. All the techniques discussed in this thesis are applicable for MA models but they do not work for ARMA model.
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BIBLIOGRAPHY. (1] H. Wold, A study in the analysis of stationary time series, Almqvist and Wiksell, Uppsala 1938. [2] Simon Haykin, Adaptive filter theory, PrenticeHall, 1986, pp. 6771. [3] Steven M. Kay, Modern spectral estimationtheory and application, PrenticeHall, 1989, p. 109. [ 4] Michael G. Larimore, John R. Treichler and C. Richard Johnson Jr, Theory and design of adaptive filters, John Wiley & Sons, Inc, 1987, pp. 4344. [5]. A. BenIsrael, A note on an iterative method for generalized inversions of matrices, Math. Comp. 20, 1966, pp. 439440. [6] Campbell + Meyer, Generalized inverses of linear transformation, Pitman publising Limited, 1979, pp. 1619. [7] see [6], pp 251255. [8] see [4], pp 91100. t9] see [4], pp 91100. [10] see [2], p 392. [ 11] D. A. Ross Constrained optimization of correlation data, private communication, 1989. [12] Alan Pankratz, Forecasting with univariate Box Jenkins models John Wiley, 1983, p. 11. [13] Ben Noble and James W. Daniel, Applied linear algebra, PrenticeHall, 1977, pp. 170173.
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71 [14] see [12] [ 15] S. Lawrence Marple, Jr, Digital spectral analysis with applications, PrenticeHall, 1987, pp. 94104. [16] see [4], pp. 267270.
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APPENDIX A This appendix contains the parameter values of the weight vector of the system, the estimates of the weight vector of the model, sum of residuals R, r.m.s of R ... of the graphs in chapters n, ill and IV. The programs are in appendix B. The computer used is Zenith.
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SINGULAR VALUK DiCOMPOSITION NUKBIR OJ PARAKITKRS : 6 KOMBIR OP DATA POINTS : 10 HU!BIR OF DATA POINTS : 20 TU ACTUAL VALOIS TBI ISTIUTIS TBI ACTUAL VALOIS TBI ISTI!&TIS OS 04999994 05 05000004 1 1 1 0 9999998 08 07999995 08 07999998 0 2 0 2 0 2 01999997 04 04000004 04 03999995 0 9 09000002 0 9 08999994 HUKBIR Ol P&RA!ITIRS : 11 NUKBIR or DATA POINTS : 10 MUKBIB OF DATA POINTS : 20 TBK ACTUAL VALOIS THE ISTI!&TIS THI ACTUAL VALOIS THI ISTI!ATKS 05 05000003 05 o5 1 0 9999993 1 0 9999999 08 07999996 08 07999991 02 02000003 02 01999995 04 04000023 04 03999993 09 09000001 09 0 8 .8000005 0 8 .7999998 .4 04000015 04 .3999998 .3 03000021 03 03000001 1 .9999984 1 1 075 07499993 .75 07500001 HO!BIR OF PAR&KITIRS : 20 MOKBIR Or DATA POINTS : 10 MOKBIR or DATA POIRTS : 20 TBI ACTUAL VALOiS TBI ISTI!ATIS TBI ACTUAL VALOIS TJI ISTIK&TIS o 5 03176904 05 05000009 1 10041876 1 09999974 08 1.058452 08 08000009 .2 08182513 02 02000022 04 01677476 04 04000015 .9 1044248 09 08999991 08 1.257589 08 07999984 04 06428237 04 03999972 3 02983337 3 .299999 1 10186358 1 09999976 075 6808423 075 07500015 105 .7937741 1 5 105 101 .6679647 101 10100001 06 01443536 06 .6000001 10 2 07956138 102 10200001 012 50106133102 012 01200018 1.55 09926724 1.55 10550001 1.21 1.109791 1.21 1.209999 066 06846545 066 06600013 109 1.171808 1.9 10899997 73
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74 SINGULAR VALUE DiCOMPOSITIOH NOMBKH OF PARAMITKRS : 6 HOKBKR OF DATA POINTS : 40 SIGHAL/NOISi : 10 THE ACTUAL VALOIS 5 1 .8 2 .4 9 SIGHAL/HOISE : 10 3 THE iSTI!ATES .496573 1.035976 .8325791 .1553249 .3893638 .9987332 THi ACTUAL VALOIS THE iSTIKATIS .5 .4996572 1 1.003597 .8 .8032578 2 .1955324 .4 .3989365 .9 .9098734 SIGMAL/HOISi : 10 TBK ACTUAL VALOIS THi ISTI!AT!S 5 4999657 1 1.000359 8 .8003258 2 .1995532 .4 .3998937 9 9009871 SIGHAL/HOISi : 100 THI ACTUAL VALOIS .5 1 .8 .2 .4 .9 SIGHAL/HOISI : 10 THE KSTI!!TiS 4989162 1. 011376 .8103024 .1858725 .3966366 .9312219 THI ACTUAL VALOIS TBI ISTIMATIS .5 .4998915 1 1.001137 .8 .8010301 2 .1985872 .4 .3996637 .9 .903122 SIGHAL/HOISI : 10 THI ACTUAL VALOIS THI ESTIMATES .5 .4999891 1 1.000113 .8 .800103 .2 .1998586 .4 .3999664 .9 .900312
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75 RECORSIYI LIAST SQOABES ITKR G( Ol G( 1) G( 21 G( 3) G( 4) G( 51 .5 1 .8 .2 .4 9 10 7746811 1.130828 1.128002 3.562172102 .471651 1.098035 50 .5452403 9960952 .8340062 .1867677 . 4260443 .9096918 100 5151973 .9967139 .8112345 .1996619 .4098652 .9010181 200 .5037134 .9995578 .8026518 .1991361 .4027318 9003244 300 5011802 9997643 .8008996 .1999186 .4008045 .9000525 400 .5004199 .9999334 .8003286 .1999284 .4003068 9000047 soo .5001542 .9999701 .8001208 600 .19997 46 . 4001105 . 9000018 .5000513 .9999892 . 8000393 700 .1999934 . 4000412 .9000002 .5000183 9999971 .8000151 800 .1999946 .4000134 .9000005 .5000066 .9999981 .8000058 .1999989 .4000034 .8999992 900 .5000021 .9999993 .800002 .1999992 .4000017 .8999996 1000 .500001 .9999999 .8000009 .1999996 .4000007 .8999999 ITKR SOH OP RISIDOALS R RHS Of R 10 3 .008534103 5. 485011102 so 3.545921105 5.954764103 100 4.051348E06 2.012796103 200 2.622408107 5.120946i:04 300 2 .689385108 1.639935104 m 3.586889109 5.989065iOS 500 4 .822382110 2 .195992105 600 5. 622787ill 7 498524106 700 7. 407871112 2. 7217 41106 800 8. 782514113 9. 311507107 900 1.155176113 3.39878807 1000 2 417282114 1.554761107
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76 RiCORSIVK LEAST SQUARES IUR 50! OF BKSIDDALS B RKS or B Gl 0) G( 1 l G( 21 G( 31 G( 4) G( Sl 1 .8 .2 .4 9 G( 61 G( 7J G( 8) G( 9) G(lOJ .8 .4 .3 1 .75 10 .3739014 .6114148 .3132842 .7471896 .5868756 .6026376 1. 949431 3.516864 .1138547 3.275992 4.25797 .3636949 0 100 1.339287104 1.157276102 5374023 .9385386 . 7251513 .148172 .399184 .95004 .7519163 .4286757 .3196768 .9856912 7417548 200 5.951337i06 2.439537i03 .50916 9874006 .7856224 .1898938 .3987489 .9085028 .7906353 U69814 .3052762 .9977764 .7502281 300 6.182849i07 7. 86311i04 .5026838 .9960766 .7952621 .1966868 .3996987 .9031326 .7970946 .4017215 .3012052 .999275 .7496806 400 7. 897627i08 2.810272104 .9985184 .7982588 .1987953 .3997553 .9009021 7991148 .4005667 .3005791 . 9996179 .7498695 500 8.681652i09 9.317539105 .5002956 .9995299 .7994212 .1995945 .3999376 .9003593 .7996843 .4001393 .3000817 .9999771 7500116 700 1. 834878110 1.354577i05 .5000549 .999935 .7999253 .1999458 .3999893 .9000444 7999479 .4000371 .3000301 .9999775 .7500006 800 l.6263U11 4.03279i06 .5000117 .9999835 .7999765 .1999824 .3999987 .9000178 .7999853 .4000045 .3000032 .9999984 .7500031 900 1. 93731112 1.391873106 .5000042 .9999939 .799992 .1999948 .3999994 .9000054 .7999946 .4000025 .30000U .9999996 7500013 1000 3. 948236E13 6.28351:07 .5000022 .999997 .7999966 .1999977 .3999992 .9000018 7999971 .4000009 .3000009 .9999987 .7499995
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77 RECURSIVE LEAST SQUARES I OF DATA POINTS = 1000 I OF ITERATIONS : 101)1) I OF PARAftETERS = b SIGNAL/NOISE RATIO = 10 ITER 61 Q) 6( 1) 6( 2) 6( 3) 6( 4) 6( 51 .5 1 .8 ., .4 .9 oL 10 7141346 1.623206 1.253718 .3473605 .8684159 1.321568 so .5554639 1.008998 .8J5n7 .1796957 .2713918 .8400678 100 .5628154 1. 085181 .8333776 .1674355 .3981438 .8587018 201) .56825 1.013933 . 7777376 .2419255 .3939645 .9182339 300 .4816526 1.047915 .8252398 .1660199 .3861295 .9101685 400 .4704792 .9298575 .7594174 .1753052 .4359998 .9638952 500 .4895214 .9352865 .7741495 .1556807 .3927307 .9340479 600 .4750061 .9602142 .8096293 .1676619 .4140025 .8831844 700 .4981464 .9094558 .8344949 .1883573 .4540443 .9572498 BOO .4886953 .9992925 .7952107 .1426926 .4640255 .9530052 900 .9905889 .8569975 .1890078 .446578 .9331438 1000 .5382441 1.036235 .8014445 .1752481 .4333977 .9247265 ITER SU" OF RESIDUALS R R"S OF R 10 .1036149 .3218926 50 2.259255E02 .1503082 100 2.75663602 .1660312 201) 2.617706E02 .1617933 300 2.81289202 .1677168 400 2.95757402 .171976 500 2.87896802 .1696752 600 2.76355702 .1662395 700 2.90953502 .1705736 BOO 2.90813902 .1705327 900 2.B32117E02 .168289 tOOO Z.B21182E02 .16796l7
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78 RICURSIVI LIAST SQUARIS I OF DATA POINTS : 1000 I OF ITIRATIONS : 1000 I OF PARAMRTIRS : 6 SIGNAL/NOISI RATIO : 10"3 ITIR Gl 0) G( 1) G( 2) G( 3) G( 4) G( 5) .5 1 .8 .2 .4 .9 10 .6259846 1.680235 1.456218 .5585368 1.124058 1.375298 50 .5195635 1.044977 .8165253 .1758852 .3916199 .8932839 100 .5130924 1. 027508 .8082988 .1858427 .4041152 .8979336 200 .5094396 1.006385 .7986541 .201(733 .4014737 .9012273 300 .4988066 1.006244 .8027991 .195801 .3989227 .9011411 400 .4972594 .9934939 .7960612 .1972776 .4037133 .9063842 500 .4990446 .9937102 .797448 .1954711 .3993407 .9033865 600 .4975378 .9960973 .800975 .196721 4014246 .8983206 700 .4998269 .9909719 .8034554 .1988193 .4054132 .905725 800 .4988737 .9999376 .7995225 .1942649 .4064055 .9052991 900 _5004745 .9990618 .8057001 .1986983 .4046593 .9033137 1000 .503825 1. 003624 .8001443 .1975238 .4033402 .9024724 nn SUM Of RISIDOALS R RKS OF R 10 .111524 .3339521 50 4 .805839104 2.192222102 100 3.496549104 1.869906102 200 2.620837104 .016189 jOO 2. 819115i04 1.679022!02 400 2.948825104 1. 717214102 500 2 .876485104 .0169602 600 2.763076104 .0166225 700 2 .909183104 1. 705633102 BOO 2.90819104 1. 705342102 900 2.832125104 1.682892102 1000 2.821194104 1. 679641102
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79 RECURSIVE LEAST SQUARES t OF DATA POINTS = 1000 t OF ITERATIONS = 1000 t OF PARAKETERS = 6 SIGNAL/NOISE RATIO = 10A5 lTER S( 0) 6( 1 I 6( 21 S( 31 S( 4) 6( 5) .5 .8 .2 .4 .9 10 .6171702 1.685939 1.476469 5796545 .149621 1.380671 50 5159735 1.048575 .814634 .1755042 .4036427 .8986057 100 .5081202 1.02174 .8057909 .4047122 .9018568 200 .5035586 .80(17459 .1974281 .4022246 300 .5005219 1.002077 .8005549 .1987i91 .4002021 .9002382 400 .4999376 .9998567 .7997253 .1994747 .4005506 9006331 500 .4999968 .9995522 .7997778 .1994501 .4000018 .9003204 600 .4997913 .9996855 .8001095 .199627 .4001668 .8998341 700 .4999951 .9991236 .8003518 .1998655 .40055 .9005722 BOO .4998914 1.000002 . 79995.J8 .1994222 .4006435 .9005283 900 .5000495 .9999096 .8005707 .1998874 .4004673 9003305 1000 .5003832 1.000363 8000145 .1997514 .4003345 .9002467 ITER SU" OF RESIDUALS RIIS OF R 10 .1138336 .337l924 5(1 3.75380304 1.937473E02 100 7.0702705 8.408489E03 200 7 .186342EOb 2.680H6E03 300 3.2477906 1.80216303 400 2.90635606 1.704B04E03 500 2.85790606 1.69053403 600 2.75928606 1. 66111E 03 700 2.9056911)6 1.70460903 BOO 2.90869406 l.705489E03 900 2.B32199E06 1.6B2914E 1000 2.821322EO& 1.679679E03
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80 &IHAUSTIVK SEARCH or PARAHETXRS = 6 or DATA= so sTxPsrzx = .1 ITIR G(O) G(l) G(2) G(3J G(4l S(51 .5 1 .:8 .2 .4 .9 1 .5059382 .9435526 .7606144 2 .2665095 .3501287 .8856738 .4898913 .9842024. .8083669 3 .2013362 ..396082 .9004196 .5007988 .9994779 . 8007211 .1996734 .3998517 .9000429 4 .500106 1.000037 .8000151 5 .1999681 .400004 .9000013 .5000042 1. 000005 .7999982 6 .1999989 .4000009 .8999999 4999999 .7999998 7 .2000001 .4 9 .5 .8000001 .2 .4 9 8 .5 .a 2 .4 .9 9 .4999999 1 .8 10 .2 .4 .9000001 5 .8 .2 .4 .9000001 ITIB SUI Of BISIDOALS R RIS Of B 1 9.821595105 9.910396103 2 3.727433106 1.930656103 3 1 .227057108 1.107726104 4 1.098729110 1.048203105 5 3.994297113 6.320045107 6 9.076767116 3.012767108 7 2.815109116 1.677829108 8 1.743744116 1.320509108 9 2 .928907116 1. 711405108 10 5.120904117 7.156049109
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81 iiB!OSTIVi SEARCH or PARA!iTXBS = 11 s or DATA= so STEPSIZI = .1 lTiR G(O) G(l) G(2) G(3) G(4) G(5) .5 1 .8 .2 .4 .9 G(6) G(7) G(8) G(9) G(lO) .8 .4 .3 1 .75 1 .5373538 .7671535 J518795 .3710172 .2474033 .8677823 .5753536 .2970887 .274225 .8567359 .6867448 2 .4582705 .9230392 .7792879 .2486658 .3506067 .9031748 .7671306 .370235 .3036527 .984106 .7401089 3 .4943438 9846756 .802288 .2061994 .3891598 9035449 .7969101 .3961909 .3013491 .9980882 7484755 4 5001471 .9975412 .8012644 .2001024 .3986763 .9007799 .7998453 .3997685 . 3003469 .9991742 .7497903 5 .5003152 .9997835 . 8002174 .1998897 .3999286 9001197 .8000241 .4000466 .300061 .999966 .7499876 6 .5000117 .999998 .8000235 .1999727 . 4000141 .9000104 .8000073 .4000138 .3000088 .9999961 .7500013 7 .5000103 1.000004 .8000005 .1999964 .4000042 .9000001 .8000013 .4000022 . 3.000009 .9999996 .7500006 8 .5000007 1.000001 .7999998 .1999997 .4000007 .8999999 .8000001 .4000003 .3000001 1 .7500001 9 .4999999 1 .8 .2 .4000001 9 .8 .4 .3 .9999999 .H00001 10 .5 1 .8 .2 .4 .9 .8 .4 .3 1 .75 ITIR SO! OJ BISIDU&LS R R!S or H 1 2. 04834103 ."0452586 2 9.515579105 9.754784103 3 3.114369106 1.781676103 4 7.441876108 2.72798104 5 1. 528291109 3.90934410S 6 4. 589051111 6.774254106 7 1.125981112 1. 061123106 8 1.516377114 1.23H13107 9 3.453488116 1.858356108 10 3.106543116 1. 762539108
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82 EIHAUSTIYE SEARCH I OF DATA POINTS = 40 I OF ITERATIONS = 10 I OF PARAftETERS = 6 SIGNAL TO NOISE RATIO = 10 ITER 6(01 6(11 6(21 6(31 6(41 &!51 .5 .8 .2 .4 .9 1 .5791272 1.100057 .9108996 5.00652602 .1510451 ,9236923 2 .4647143 1.078307 .7706338 4.498323E02 .3231962 .9823988 3 .5030513 1.025966 .7840217 9.528696E02 .3584524 1.001091 4 .5203474 1.020781 .7918397 .1086689 .3681998 1.004547 5 .5236301 1.020338 .7943437 .1121429 .3702465 1.005102 6 .5242792 1.020436 .7950426 .1128936 .3706305 1.005157 7 .5243801 1.020503 .7952063 .113036 .3706873 1.005152 8 .5243895 1.020526 .79524 .UJ0576 .3706916 1.005147 9 .5243883 1.020532 .795246 .1130596 .37069011 1.005145 10 .5243871 1.020534 .7952468 .1130592 .3706898 1.005145 ITER SUft OF RESIDUALS R RRS OF R STEPS lZE = .1 1 1 2.115553702 .1629582 2 1.88865402 .1374283 3 1.80185602 .1342332 4 1.79507902 .1339805 5 1. 794 738E 02 .13391178 6 1.794722E02 .1339672 7 1.79472202 .1339.72 8 1.79472102 .1339672 9 1.79472202 .1339672 10 1. 79472102 .1339672
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83 EXHAUSTIVE SEARCH I OF DATA POINTS = 40 I OF ITERATIONS = 10 I OF PARA,.ETERS = 6 SIGNAL TO NOISE RATIO = 10A3 ITER G!OI G(11 6(2) 6(31 6(4) 6(51 .s 1 .8 .2 .4 .9 .5476148 1.067056 .8826056 2.41371902 .1925286 .8378168 2 .447068 1.050989 .7726919 .1272363 .3537032 .8907127 3 .4834038 1.006088 .7886431 .1750563 .3862042 .9071142 4 .4989363 1.002076 .7963551 .1873996 .39492 .9100473 5 .5018057 1.001835 .7987092 .190512'l .3967018 .9104'J32 6 .5023555 1.001958 .7993452 .1911683 .3970246 .'ll05294 7 .502436 1.002025 . 7994901 .1912885 .397069 .'l10522 8 .5024418 1.002046 . 7995193 .191J056 .3970712 .9105169 9 .5024401 1.002052 7995241 .1913066 .3970699 .910515 10 .5024391 1.002053 .7995248 .1913062 .3970692 .9105146 ITER ITER SU" OF RESIDUALS R R"S OF R 1 7.2602103 8.52068602 2 9.711315E04 3.1162'l8E02 l 2.38034904 1.54283802 4 1.823075E04 1.35021302 5 1.79603704 1.34016302 6 l.794774E04 1.33969202 7 1. 794 722E 04 1.JJ9672E02 8 l.794721E04 1.33967202 9 1. 794 722E 04 1. 33967302 10 1. 79471904 1.3396 71E 02
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84 EXHAUSTIVE SEARCH I OF DATA POINTS = 40 I OF ITERATIONS = 10 t OF PARA"ETERS = 6 SIGNAL TO NOISE RATIO = ITER 6(0) G(l) 6(2) 6(3) 6(41 6(5) .s 1 .8 .2 .4 .9 1 .5444685 1.063756 .8797762 3.155838E02 .1966778 .8292298 2 .4453036 1.048258 .7728978 .1354619 .3567541 .8815441 3 4814392 1. 0041 .7891053 .1830333 .3889794 .8977166 4 .4967953 1.000206 .7968065 .1952727 .397592 .9005974 5 .4996233 .9999848 .7991458 .1983499 .3993473 .9010324 6 .5001632 1.000ll . 7997753 .1989958 .3996641 .9010668 7 .5002416 1.000177 .7999186 .1991137 .3997072 .901059 B 5002471 1.000198 .7999472 .1991304 .3997092 .9010538 9 .5002453 1.000204 .799952 .1991313 .3997079 .9010521 10 .5002443 1.000205 .7999525 .1991309 .3997072 .9010516 ITER SU" OF RESIDUALS R R"S OF R STEPSIZE = .1 1 b.946154E03 B.334359E02 2 7. 797317E 04 2.792368E02 3 5.916066E05 7.691597EOJ 4 4.56192E06 2.135865E03 5 1.922627E06 1. J8b58BE OJ b 1.799b4BE06 1.34151E03 7 1. 794914E06 1.J39744EOJ B 1.7'l4741EOb 1.JJ967'lEOJ 9 1. 79473'lE 06 l.JJ9679E03 10 1. 794714E 06 1.3J9669E03
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APPENDIXB PROGRAM LISTINGS All the programs in this appendix are written in GWBASIC language. program "Singular value decomposition" is from [15] it was written in FORTRAN and translated into GWBASIC. *program "Recursive least squares is from [16], it was also written in FORTRAN and translated into GW BASIC. the major and main part of program "Exhaustive search algorithm" : subroutine parameters search" was developed by professor Douglas 'A. Ross [ 11].
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10 . 20 '11111111111111111 SINGULAR VALUE DECDftPOSITION 111111111111111111111 30 . 40 'DEFDBL O,M,E,R,T,Y,U,O,P,A,S,D,F,G,H,Z,l,C,V,B SO ft=30 : L=20: ftftft=ftL 60 Dlft llfti,Difti16ILI,YIL,1l 70 FOR TO ft1 80 l(K)=21(RND.SIISGR(Jt.1l 90 NEXT K 100 610 1=.5 : 6(1 1=1 110 6(4 1=.4 : 6(5 1=.9 120 6(8 1=.3 : 6(9 )=1 130 61121=1.1 : 6(13)=.6 140 6(161=1.55 : 6(17)=1.21 150 FOR K=L1 TO ft1 :6(2 ) = .8 :Sib l =.8 :6(10) =.75 :6(141 =1.2 :6(19) =.66 :6(J 1=.2 :6(7 )=.4 :6(11)=1.5 :6(15) :.12 :6( 19)=1. 9 160 DIKI= 6(0)1l(K)+6(1)11(K11+6(211l(K2)+6(3)11(K3)+6141ti(K41 170 DIKI= DIKI+61511l(K51+6161tX(K61+G(7111(K71+61BIII(K81 180 Dill= DIKI+6(9)tl(K91+6(1011IIK101+6(11)11(K111+6(12111(K121 190 DIKI= DIKI+6(131111K131+6(14)1l(K14)+6(1SIII(K15)+6(1611l(K161 200 DIKI= 210 NEXT K 220 '111111111111111111111 COftPUTE R & P 11111111111111111111111111111111 230 Dift P(L1111,RIL1,L11,AIL,L),ASIL,LI 240 KK=ft1 250 FOR I=O TO L1 260 T=O 270 FOR J=L1 TO KK 280 T=T+D(J)Il(J11 290 NEXT J JOO PII11l=T : Y(I+1,1)=P(I,1) 310 NEXT I 320 FOR I=O TO L1 lJO FOR J=l TO L1 340 T=O 350 FOR Jl=tl111 TO IKK11 l60 T=T+l(Jl)ll(J1J+Il l70 NEXT Jl 380 RII,JI=T : R!J,II=RII,Jl J90 AII+1,J+1l=RII1J) : A(J+l11+1)= A(I+11J+ll 400 NEll J 410 NEll I 420 11111111111111&1&1 PRINT P AND R 11111111111111&111111111111&111111 430 PRINT 1 THE ELEftENTS DF (L1)11 ftATAil PARE (l ELEftENTS): 440 PRINT THE ELEftENT DF ftATRil p STARTS (0,1) I 450 FOR 1=0 TO L1 460 PRINT Ptl,11 470 PRINT 480 NEXT I 490 PRINT THE ELEftENTS OF ftATRil (L1)1(L1) RARE (lll ELEftENTS): 86
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500 PRINT THE ELEftENT OF ftATRII R STARTS (010) 510 FOR 1=0 TO L1 520 FOR J=O TO L1 530 PRINT R(I1Jl1 540 NEIT J 550 PRINT 560 NEXT I 570 .ENTER A1ft1N1NU,NV1S1U1V11P 580 aaaaatllllllllllt SINGULAR VALUE DECOftPOSITION lllllllllltlllllllllll 590 'llllflllflllllllllllllll A=UISIY(H) ttlllllllllllllllllllllllllllllll 600 . A="'" ftATRII I U="'" I S=ftiN I V= NIH 610 ft=L : N=L : NU=ft : NV=N : IP=O 620 "ftAX=" : NftAI=N 630 Dlft U(ft1ft)1V(N1N)1UH(ft1")1VH(N1N)15"1"1N)11Sft(N1ft) 640 Dl" SINJ,B(l00)1C(100)1T(100) 650 ETA= 1.207 : TOL= 2.4El2 660 NP=N+IP :Nl=N+l 670 FOR 1=1 TO ft 680 FOR J=1 TO N 690 A5(11J)= A(I,JI 700 NEXT J 710 NEll I 720 730 740 750 760 770 780 'lllllllllllltllllllllll HOUSEHOLD REDUCTION tllllllllllllllllllllllll Clli=O K=l Kl=K+l 'lllllllllllllllll EllftiNATION OF A(l1k)1I=K+l1 1 111111111111111 Z=O FOR I=K TO ft 790 Z=Z+A(I1K)A2 BOO NEXT I 810 9(1)=0 820 IF Z<=TOL GOTO 1060 BJO Z=SDRill 840 BIKl=Z 850 W=AB5(A(k1K)) 860 0=1 870 IF W>
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990 NEIT J 1000 '8111tllllllltllllllll PHASE TRANSFORftATION tllllllttlllllllltltllltl 1010 0=A(K,K)/ABS(A(K,KII 1020 FOR J=K1 TO NP 1030 A(K,J)=QIA(K,J) 1040 NEXT J 1050 'tllltllttlttlllll ELI"INATION OF A(K.J)1J=K+21 ,,, 1N lllltlttlllltl 1060 IF K=N 60TO 1370 1070 l=O 1080 FOR J=K1 TO N 1090 1100 NEXT J 1110 C(ll 1=0 1120 IF 1<= TOL 60TO 1340 1130 Z=SORIZI 1140 C(Ul=Z 1150 M=ABS(A(K,K1ll 11&0 0=1 1170 IF W>EPS THEN EPS=S(K)+T(K) 1420 NEH K 1430 EPS=EPStETA 1440 '11111111111111111111 INITIALIZATION OF U AND V llllllllllllllttlttllltt 1450 IF NU=O 60TO 1520 14&0 FOR J=l TO NU 1470 FOR 1=1 TO 88
PAGE 100
1480 U(I,J)=O 1490 NEXT I 1500 U(J,Jl=1 1510 NEXT J 1520 IF NY=O 6DTO 1600 1530 FOR J=1 TO NY 1540 FOR 1=1 TO N 1SSO YII,J)=O 1560 NEIT I 1570 YIJ,J1=1 1580 NEXT J 1590 '111111111111111111111 DR DIASONALIZATION 1111111111111111111111111111 1600 FOR KK=l TO N lla10 K=NHK 1620 'llllllllltllllllllllll TEST FOR SPILT 111111111111111111111111111111 1630 FOR LL=1 TO K 1640 L=K+1LL 1650 IF ABS(TILll<=EPS THEN GOTO 1970 1660 IF A8S(S(L1ll<=EPS GOTD 1690 1670 NEIT LL 1690 '11111111111111111111111 CANCELATION OF BILl lflllllfflfllllllllllllll 1690 CS=O 1700 SN=1 1710 ll=L1 1720 FOR I=L TO K 1730 F :oSNU II I 1740 TII)=CSIT(I) 1750 IF ABSIF)<=EPS SOTO 1970 1760 H=SIIl 1770 W=SDR(FA2+HA2) 1780 5(1)=16 1790 CS=H/11 1900 SN=F/N 1810 IF NU=O SOTO 1990 1920 FOR J=1 TO N 1830 I=U(J,lll 1840 Y=UIJ,I) 1850 UIJ,Ll)=IICS+YISN 1860 U(J,I)=YICSIISN 1970 NEXT J 1880 IF NP=N GOTO 1950 1890 FOR J=Nl TO NP 1900 O=Aill,J) 1910 R=AII,J) 1920 AIL1,J)=OICS+RISN 1930 AII,J)=RICSOISN 1940 NEXT J 1950 NEIT I 1960 '1111111111 TEST FOR CONVERGENCE llllllllllllllllfllllllllllllllllfllll 89
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1970 II=S(q 1980 IF L=K SOTO 2580 1990 '11111111111111111111111 ORIGIN SHIFT llttllttlll11tlltlllllllllllttll 2000 l=Sill 2010 Y=S(K11 2020 G=T( l1 I 2030 H=T(K) 2040 F=((YII)I(V+II)+(6H)t(6+H)l/(21HIY) 2050 2060 IF F
PAGE 102
2460 FOR J=N1 TO NP 24 70 Q=A ( 11, J ) 2480 R=AII, J) 2490 AII11J)=QtCSRISN 2500. A(I,Jl=RICSGISN 2510 NEIT J 2520 NEXT I 2530 l(Ll=O 2540 T(Kl=F 2550 SIKl=l 2560 GOlD 1630 2570 'lllllllllllllllllllllll CONVERGENCE lllllllllllllllllllllllllllllllll 2580 IF W>=O GDTO 2640 2590 S!Kl=11 2600 IF NY=O SOlO 2640 2610 FOR J=l TO N 2620 VIJ,K)=VIJ,Kl 2630 NEXT J 2640 NEXT U 2650 '1111111111111111111 SORT SINGULAR VALUES llllllllllllllllllllllllllll 2660 FOR k=l TO N 2670 6=l 2680 J=K 2690 FOR l=l TO N 2700 IF 5111<=6 GOTO 2730 2710 6=5(1) 2720 J=l 2730 NEXT I 2740 IF J=l 60TO 2950 2750 S(J)=SI ll 2760 SIKl=& 2770 IF NV=O 60TO 2830 2780 FDA 1=1 TO N 2790 O=VII I J l 2800 VII,J.l=Y( I ,ll 2810 YII,Kl=O 2820 NEXT I 2830 IF NU=O SOlO 2890 2840 FDA 1=1 TO N 2850 Q=U!I,Jl 2860 UII,Jl=Uti,Kl 2870 U(I1K)=Q 2880 NEXT I 2890 IF N=NP 60TO 2950 2900 FDA I=Nl TD NP 2910 Q=A(J11) 2920 AIJ,II= A(K,II 2930 A( K, ll=Q 2940 NEXT I 91
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2950 NEXT l 2960 tttttlllltta BACK TRANSFORftATIDN lltlllllltttlllltlltlltttlllllllllll 2970 IF NU=O GOTO 3160 2980 FOR KK=l TO N 2990 K=N1U 3000 IF BIKI=O GOTO 3150 3010 0=AIK,K)/ABSIAIK,Kll 3020 FOR J=1 TO NU 3030 U(K,J)=OIU(K,Jl 3040 NEXT J 3050 FOR J=1 TO NU 3060 0=0 3070 FOR I =K TO II. 3080 Q=O+A(I,K)tU(I,JJ 3090 NEXT I 3100 0=0/ ABS(A(l,KiiBil)) 3110 FOR I=K TO II 3120 U(I,J)=U(I,J)OIA(I,Kl 3130 NEXT I 3140 NEXT J 3150 NEXT U 3160 IF NV=O GOTO 3510 3170 IF N<2 60TO 3510 3180 FOR KK=2 TO N 3190 K=N1U 3200 U=K+1 3210 IF C(K1J=O GDTO 3360 3220 O=AIK,K11/A8S(AIK,K1ll 3230 FOR J=l TO NY 3240 V(K11J)=OIV(K1,J) 3250 NEXT J 3260 FOR J=l TO NV 3270 0=0 3280 FOR l=l1 TO N 3290 O=O+A(K,l)IV(l,Jl 3300 NEXT I 3310 O=O/ABSIAIK,K1)tCIK1ll 3320 FOR I=Kl TO N 3330 YII,Jl=V(I,JJDIA(K,II 3340 NEXT I 3350 NEXT J 33110 NEIT KK 3370 PRINT"USE SINGULAR VALUE DECOIIPOSITION TO SOLVE:" 3380 PRINT "R(LILI I M(Lil) = P(llll IIATRII EQUATION" 3390 PRINT "LET R(LIL)=AS(LtL) P(Lil) = Y(Lil)" 3400 PRINT"USE SINGULAR VALUE DECOftPDSITION TO SOLVE:" 3410 PRINT "ASilll) I W(lll) = Y(Lil) ftATRII EQUATION" 3420 PRINT AS=UI SIIIV(Hl l4l0 PRINT U(H)IASIY=SII 92
PAGE 104
3440 PRINT W=YIIS"IUIHIIY 3450 PRINT U"'" UIHER"ITIANl"'"" 34&0 PRINT AS"IN" 3470 PRINT YNIN, YIHEAftiTIANININ" 3480 PRINT S""IN, IS" Nift" 3490 PRINT THE SINGULAR VALUE DECOftPOSITION OF AS KIN ftATRII" 3500 PRINT I U(HERftlTIANIIASIY=S"" 3510 '111111111111111111 PRINT SINGULAR VALUES 1111111111111111111111111111 3520 PRINT "THE SINGULAR VALUES ARE:" 3530 FOR 1=1 TO N 3540 PRINT Sill, 3550 NEll I 3560 PRINT 3570 '11111111111111111111 "ATRIX UIHEA"ITIANI 11111111111111111111111111111 3580 FOR 1=1 TO 3590 FOR J=l TO ft 3600 UHII,JJ=UIJ,I) 3610 NEXT J 3620 NEXT 1 3630 '11111111111111111111 PRINT THE "ATRIX U lttllllllllllllllllllllllllll 3640 PRINT "THE ELE"ENTS OF "'" ftATRIX U ARE:" 3650 FOR l=l TO 3660 FOR J=1 TO 3670 PRINT U(I,J), 3680 NEXT J 3690 PRINT 3700 NEXT I 3710 '11111111111111111111 PAINT THE "ATRII UIHERftlTIANI 1111111111111111111 3720 PRINT "THE ELE"ENTS OF "'" "ATAII UIHER"ITIANI ARE:" 3730 FOR 1=1 TO 3740 FOR J=l TO 3750 PRINT UH(I,Jl, 3760 NEXT J 3770 PRINT 3780 NEXT I 3790 PRINT THE KATAIX AS llltlallllllllllaaaalllllllll 3800 PRINT "THE ELEftENTS OF KIN ftATRIX AS ARE:" 3810 FOR 1=1 TO K 3820 FOR J=1 TO N 3830 PRINT ASII,Jl, 3840 NEXT J 3850 PRINT 3860 NEXT I 3870 '11111111111111111111 KATRII VIHER"ITIAN) tllllllllllllllllllltltllllltl 3880 FDA TO N 3890 FOR J=l TO N 3900 YHI!,J)=YIJ,Il 3910 NEIT J l920 NEXT I 93
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3930 'lllllllltlllllllllll PRINT THE IIATRIX V lllllllltllitlllllllllllllltll 3940 PRINT 'THE ELEIIENTS OF NIN IIATRII Y ARE:" 3950 FOR 1=1 TO N 3960 FOR J=1 TO N 3970 PRINT V(I,JI, 3990 NEIT J 3990 PRINT 4000 NEXT I 4010 'llllttllllllllllllll PRINT THE ftATRII VIHERftiTIANI 1111111111111111111 4020 PRINT 'THE ELEftENTS OF NIN IIATRII V(HERIIITIANJ ARE:' 4030 FOR 1=1 TO N 4040 FOR J=1 TO N 4050 PRINT VH(I,JI, 4060 NEIT J 4070 PRINT 4090 NEXT I 4090 'llllllllllltllllllllllllllll IIATRII Sft llllllllllllllllllllltllllllll 4100 FOR 1=1 TO ft 4110 FOR J=1 TO N 4120 SIIII,J I =0 4130 IF I=J THEN SII(I,JJ=S(I) 4140 NEIT J NEXT I 4160 '11111111111111111111 PRINT THE 511 11111111111111111111181111111 4170 PRINT "THE ELEIIENTS OF ftiN ftATRIX Sll ARE:" 4190 FOR I=1 TO II 4190 FOR J=1 TO N 4200 PRINT SII(I,JI, 4210 NEXT J 4220 PRINT 4230 NEXT I 4240 '181111181811111111111111111 IIATRIX 1511 1111111111111111111111111811111 4250 FOR 1=1 TO N 4260 FOR J=1 TO II 4270 1511(1,JI=O 4290 IF I=J AND 5(11> .0001 THEN IS111l,JJ=1/S(II 4290 NEXT J 4300 NEXT I 4310 IIIIUUIIIUIIUIU PRINT THE ftATRII 1511 UIUUUIIIIIUIUUIIUIUI 4320 PRINT 'THE ELEftENTS OF Nlll IIATRIX 1511 ARE:' 4330 FOR I=1 TO N 4340 FOR J=l TO II 43SO PRINT ISft(I,JI, 4360 NEXT J 4370 PRINT 4380 NEXT I 4390 Dill VIIN,II), YIU(N,III, US(II,N), USYIII,NJ,W(II,11 4400 'Dlft VI(S,Sl, VIU(S,Sl, US(S,Sl, USV(S,Sl,W(S,ll 4410 PRINT'USE SINGULAR VALUE DECOftPOSITION TO SOLVE:' 94
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4420 PRINT "RIULI I II(LI1) = Plllll IIATRIX EQUATION" 4430 PRINT "LET R(LIL)=AS(LIL) I P(Lill = Y!Lil)" 4440 PRINT"USE SINGULAR VALUE DECOIIPOSITION TO SOLVE:" 4450 PRINT "AS(LIL) I ll(llll = Y(Lil) IIATRIX EQUATION" 4460 PRINT AS=UI SftiV(H) 4470 PRINT U(H)IAStY=SII 4480 PRINT II=YfiSIIIU(HlfY 4490 PRINT U111111 UIHERIIITIAN)11111" 4500 PRINT ASIIIN" 4510 PRINT YNIN, YIHEAIIITIAN)NIN" 4520 PRINT I SllIUN, ISII NIII" 45JO FOR I=l TO N 4540 FOR J=l TO II 4550 YIII,J)=O 4560 FOR K=l TO N 4570 Vl(l,J)=YI(I,J)+Y(l,K)IISIIIK,Jl 4580 NEXT r 4590 NEXT J 4600 NEXT I 4ol0 FOR 1=1 TO N 4620 FOR J=l TO II 4630 YIU(I,J )=0 4640 FOR K=1 TO II 4650 YIUII,Jl=VIUII,J)+VI(l,KIIUH(K,Jl 4660 NEXT k 4670 NEXT J 4690 NEXT I 4690 FOR 1=1 TO N 4700 FOR J=l TO 1 4710 ltii,JI=O 4720 FOR K=l TO II 4730 llll,J)=II(l,J)+VIUII,K)IYIK,JI 4740 NEXT K NEll J 4760 NEXT I 4770 'lllfflllllllllllfllf PAINT THE IIATRII Yf!SIIIUIHI 1111111111111111111 4780 PAINT "THE ELEIIENTS OF Nlll IIATAII YIISIIIU(H) ARE:" 4790 FOR 1=1 TO N 4900 FOR J=1 TO II 4810 PRINT YIU!I,J), 4820 NEXT. J 4830 PRINT 4840 NEJT I 4850 'IIIIIUUUIIU CHECK AS =UI SillY IUIIUUIIIUUIIIIUUUIIUUUI 4860 PRINT CHECK AS =UI SillY 4870 FOR 1=1 TO II 4890 FOR J=l TO N 4890 US( I,J l=O 4900 FOR K=l TO II 95
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4910 USII,Jl=US(I,Jl+UII,llf Sft(K,Jl 4920 NEll K 4930 NEXT J 4940 NUT I 4950 FOR 1=1 TO ft FOR J=1 TO N 4970 USVII,J)=O 4980 FOR K=1 TO N 4990 USV(I,Jl=USVII,J)+US(I,KIIYHIK,JI 5000 NEll K 5010 NEXT J 5020 NEXT I SOlO aalllllllltlllllllll PRINT THE ftATAII AS=UI SftiY(H) 11111111111111111111 5040 PRINT "THE ELEftENTS OF ftiN ftATRII AS=UI SftiYIHJ ARE: 5050 FOR I=l TO 50&0 FOR J=l TO N 5070 PRINT USY(I,JJ, 5080 NEll J 5090 PAINT 5100 NEXT I 5110 PRINT COftPUTE N=YIISfttU(H)IY 5120 FOR I=1 TO N 5110 W(l ;1)=0 5140. FDA K=1 TOft 5150 W(l,ll= W!I,ll+YIUII,Kll Y(K,ll 51&0 NEIT K 5170 NEIT I 5180 LPRINT TAB!Ol;"NUftBER OF DATA POINTS= ";ftftft 5190 LPRINT TABIOJ;"THE ACTUAL YALUES";TA8(20l;"THE ESTI"ATES" 5200 FOR 1=1 TO N 5210 LPRINT TAB!OJ;6(11J;TABI20l;ii(I,ll 5220 NEXT I 5230 END 96
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10 'lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll 20 'lllllllllllltltfllf RECURSIVE lEAST SQUARES tllllllltltllltllllll 30 ltllllllllllltlllllllllllllllllllllllllllllllllll KEY OFF:CLS 50 'DEFDBl J,M,G,R,D,C 60 'liNREAl INPUT SAftPlE AT TlftE K 70 'YDUTAEAl OUTPUT SAftPlE AT TlftE K 80 'DDESIRED OUTPUT SAftPLE AT TlftE K 90 'RHOREAl EXPONENTIAL FORGETTING FACTOR 100 'MARRAY CONTAINING MEI&HT VECTOR AT TlftE K 110 'NORDNURSER OF TAPS IN THE FILTER 120 'ITDLARRAY FOR STORAGE OF PAST DATA SAftPLES 130 'IPTRPOINTER FOR DELAY LINE ARRAY 140 'RINVREAL ARRAY FOR STORAGE OF INVERSE CORRELATION ftATRil 150 60SU8 250 : RE" DI"ENSIONS AND INITIALIZATION 160 PRINT PLOT ESTiftSTES OR RESIDUALS DR OUTPUT E/R/0 ' 170 INPUT AS. 180 IF AS="O" SOTO 230 190 IF AS="E" SOTO 210 200 IF AS="R" SOTO 220 210 GOSUB 1430 : SOTO 230 220 GOSUB 1780 230 SOSUB 430 : REft RECURSIVE LEAST SOUARES ALGORITHft END 250 REft llltlltlttll DlftENSIONS AND INITIALIZATION ttlttlltllllltllll 260 11=0 : 12=0 270 ft=956 :C=S :l=S :16PTR=L :ITER=O :EO=O :RH0=.99 280 Dift Xlfti,ITDLIL+11,Nil+11,6(25 l,Z(l+l),RINV((Lt11A21,Difti,C!fti,Y1(L+11 290 FOR 1=0 TO l 300 XTDL (I 1=0 310 N!II=O : Y1(11=0 320 RINV(II(l+11+11=5000 no NEXT I FOR 12 = o TO l J50 'RP(0112J=O : EP!O,I21 = 0 360 NEIT 12 370 6(01= .5 :6(1)= 1 :G(21= .8 :6(31= .2 :6(4)= J80 6!51= .9 390 LPRINT TAB!OI;"ITER' 400 lPRINT TA8(01;"6( O)'jTA8(141;'6( 11';TA8(281;"6( 2J";TA8(42J; "&( 31"; TAB!561 ;"6( 41";TA8(701 ;"G! 5)" 410 lPRINT TAB(OI; 61 OJ ;TA8(14J; 61 11 ;TA8128J; 61 21 ;TA8142J; 6( 31 ; TA8(56) ; G( 41 ;TA8(701 i 6( 51 420 RETURN 430 REft llllltltltlllf RECURSIVE lEAST SOUARES ALGDRITHft llfllllllflfft 440 'Ill INPUT' DESIRED OUTPUT Ill 450 REft D!K) = G(O)Il(KI+ G(L)ti(KLl 460 FOR k=O TO ft1 97
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470 I!Kl=2t!RND.5)t5DR!3t.01) 480 NEXT K 490 IIN=X(IGPTR) 500 ACCUII=O 510 LPTA=ISPTR 520 FOR 1=0 TO L 530 ACCUII=ACCUII+SIII&l(LPTRl 540 LPTR=LPTR1 550 NEXT I 5b0 16PTA=I6PTA+1 570 IF ISPTA 1 = 111 GOTD 240 : 111!Ll = I OF ITERATIONS OF RLS 580 D=ACCUII 590 'Ill ESTIIIATE OUTPUT Y(Kl Itt bOO ITOL(IPTR)=l!N blO ACCUII=O &20 LPTR=IPTR 630 FOR 1=0 TO L b40 ACCUII=ACCUII+W(I)IITDL!LPTRI &SO LPTR=LPTR+l bbO IF LPTR >L THEN LPTR=O 670 NEXT I 680 YOUT=ACCUII 690 'Ill COIIPUTE ERROR &It 700 E =DYOUT 710 'Ill COIIPUTE FILTERED INFORIIATION VECTOR Z ttl 720 FOR 1=0 TO l 730 LPTR=IPTR 740 ACCUII=O 750 FOR J=O TO l 7b0 ACCUII =ACCUII+RINV(JI(L+1)+1)11TDLILPTR) 770 LPTR=LPTR+l 780 IF LPTR >L THEN LPTR=O 790 NEXT J Z!I)=ACCUII 810 NEIT I 820 'til COIIPUTE O=I(TRANSitZ til 830 ACCUII=O 840 LPTR=IPTR 850 FOR 1=0 TO l 860 ACCUII=ACCUII+ITDLILPTR)IZIII 870 LPTR=LPTR+l 880 IF LPTR > L THEN lPTA=O 890 NEll I 900 O=ACCUII 910 aaa COI!PUTE GAIN CONSTANT tat 920 Y=l/IAHO+Q) 930 'Ill UPDATE WEIGHTING COEFFICIENTS ttl 940 FOR 1=0 TO L 950 W(II=W!II+EIVIZ(II 98
PAGE 110
960 NEXT I 970 'Ill UPDATE R INVERSE ftATRIX Ill 980 FOR 1=0 TO L 990 FOR J=I TO L 1000 JPTR= JIIL+1)+I 1010 KPTR= II(L+1)+J 1020 RINV(JPTRI=IRINV(JPTRJZIIIIZ(JJIVI/RHO 1030 RINYIKPTRJ=RINV(JPTRI 1040 NEXT J 1050 NEXT I 1060 IPTR=IPTR1 1070 IF IPTR < 0 THEN IPTR=L 1080 ITER = ITER+1:60SUB 1290 :ITER = ITER 1 1090 IF AI="E" 6010 1120 1100 IF AS="R" GOTO 1130 1110 ITER=ITER+1 :12=ITER : 60SUB 1150 : 60TO 490 1120 ITER=ITER+1 :12=ITER : &OSUB 1690 : &OTO 490 1130 ITER=ITER+1 :16=1TER : 60SUB 1890 : 60TO 490 1140 REft 111111111111111111 PRINT RESULTS 111111111111111111111111111111 1150 CLS 1160 LPRINT 1170 LPRINT TABIOI; ITER 1180 LPRINT TABIO); Ml OJ ;TABI141; Ml 11 ;TAB12BI; M( 2) ;TA8142); Ml 3) ; TABI56J ; WI 41 ;TA8(70) i M( 5J 1190 LOCATE 1 ,1 :PRINT "6(0 1=";610 ):LOCATE 2 ,!:PRINT Ml 0 I 1200 LOCATE 1 ,25:PRINT "611 1=";6(1 ):LOCATE 2 ,25:PRINT Ml 1 I 1210 LOCATE 1 ,SO:PRINT "6( 21=";61 2J:LOCATE 2 ,50:PRINT WI 2 I 1220 LOCATE 3 ,1 :PRINT "6(3 1=";613 ):LOCATE 4 ,1:PRINT Ml J I 1230 LOCATE 3 ,25:PRINT "614 1=";61 4J:LDCATE 4 ,25:PRINT Nl 4 I 1240 LOCATE 3 ,SO:PRINT "6( 51=";615 ):LOCATE 4 ,50:PRINT MIS I 1250 LOCATE 18,25:PRINT "ITER =";ITER 1260 LOCATE 19,25:PRINT "SUft Of RESIDUALS R : ;R 1270 LOCATE 20,25:PRINT "RftS RESIDUALS = SORIRI = ";SDRIRI 1280 RETURN 1290 REft 11111111&11111111111111 COftPUTE RESIDUALS 111111111111111111111 1300 R=O : 01=0 : U2=l 1310 FOR I=L TO l+ITER 1320 03=02 1330 Dlll=O : Clll=O 1340 FOR J=O TO L 1350 Dill= DIIJ+ 61JI11(03J 1360 Clll= CIIJ+ M(JJIXIOJJ 1370 03=031 :NEIT J 1380 02=02+1 1390 1400 NEXT I 1410 R=R/IITER+ll 99
PAGE 111
1420 RETURN 1430 REft lttttlllltllttttltt lY PLANE FOR ESTiftATES ltltltttttltllttltt 1440 W=2 1450 CLS : SCREEN 2 1460 FOR Y=O TO 160 STEP 20:LINE (38,Y 1(40,Y I : NEIT Y 1470 FOR Y=39 TO 589 STEP 110:LINE (Y,79J(Y1811 : NEIT Y 1480 FOR 1=1 TO S :LOCATE 111141I+1 :PRINT (ft1LIII/S: NEXT I 1490 LOCATE 1,1 : PRINT W : LOCATE 20,1 :PRINT M 1500 LOCATE 5,1 : PRINT W/2 : LOCATE 16,1 :PRINT W/2 1510 LINE (39,0)(39,160) 1520 LINE (39,80 1,80 1530 FOR 1=0 TO L 1540 X=6(1) 1550 60SUB 1580 1560 NEXT I 1570 RETURN 1580 IF 1<0 60TO 1640 1590 Z= 80 (Xt199)/(M+M+MI39/801 1600 FOR J=l9t1000/600 TO 600 STEP 25 1610 PSETIJ,ZI 1620 NEIT J 1630 &OTO 1680 1640 l=(W+ABS(I))t199/(W+W+Mtl9/80) 1650 FOR K=l911000/600 TO 600 STEP 25 1660 PSETIK,ZI 1670 NEXT I 1680 RETURN 1690 REft tttlttttltttllttllllt PLOT THE ESTiftATES ttttttltllttttttttltlt 1700 VIEW(39,0)(589,160) 1710 NINDOW(O,N)(ftL1M) 1720 LOCATE 22,70 : PRINT "ITER=";ITER 1730 FOR l=O TO L 1740 LINE (11,Y1(0))(121M(01) :LINE. IIl,Y1(1))(12,Milll 1750 NEIT I 1760 FOR 1=0 TO L : Y1(l)=Nill : NEIT I : X1=12 1770 RETURN 1780 REft ltttttttttttlttttt XV PLANE FOR RESIDUALS ttttllttltlltltltt 1790 CLS : SCREEN 2 1800 LINE 1119,8 )(119,136 I 18IO LINE (1I9,136 115201136 I 1820 FOR Y=8 TO 136 STEP 8 :LINE 111B,Y )(120,Y I : NEIT Y 1830 FOR Y=119 TO 520 STEP (520119)/IC11 :LINE (Y,1l7 )(Y1135): NEIT Y 1840 LOCATE I9,I5:PRINT !:LOCATE 19,65: PRINT ft1L 1850 A= 9 :8= 1 :NIN=10A(A) : ftAI= 10AIBI:K=IABSIAJABS(8JJ/16 1860 FOR I=O TO 16 STEP 4 :LOCATE 1+2,1 :H=III+1 :PRINT "10E"(HJ :NEXT I 1870 FOR 1=0 TO 16 STEP 4 :LOCATE 1+2,11 :H=Itl+l :PRINT (HI :NEXT I 1880 RETURN 1890 REft tltttttttllltltttltlltt PLOT RESIDUALS ttllllllttttllttttlttlt 1900 A= 9 :8= 1 100
PAGE 112
1910 'R=L061Alll06110l 1920 LOCATE 20,3 : PRINT 'R':LOCATE 20,9:PRINT "L06(AI' 1930 LOCATE 20,20: PAINT 'SU" OF RESIDUALS A = ;R 1940 LOCATE 21,20: PRINT 'R"S OF R = ';SOR(RI 1950 LOCATE 22,20: PAINT'LOS!R) = '; L06!AJ/L06(101 19b0 LOCATE 23,20: PRINT'ITER = ';ITER 1970 Ab=L06(Al/L06(101 1980 VIEW1119,B )(520,136) 1990 WINDOW !1, B )("1L, A I 2000 IF 15=0 6010 2020 2010 LINE(151R5)(Ib,Abl 2020 15=1b : RS=Rb 2030 RETURN 101
PAGE 113
10 20 '1111111111111111111111 EXHAUSTfYE SEARCH AL60RITH" 11111111111111 30 40 KEY OFF:CLS SO 'DEFDBL X,W,6,R,C,D 60 60SUB 180 : REft DlftENSIONS AND INITIALIZATION 70 PRINT"PLDT ESTI"ATES OR RESIDUALS OUTPUT E/R/0 SO INPUT AS 90 IF At="O" SOTO 130 100 IF AS="E" SOTO 120 110 GOSUS 1310:6010 130 120 GDSUB 940 130 GOTO 160 140 GOSUB 460 : AE" EXHAUSTIVE SEARCH 150 END 160 GOSUB 880 : REft Ill GENERATED INPUTS AND OUTPUTS FOR ES 170 SOTO 140 180 HEft 111111111111 DiftENSIONS AND INITIALIZATION lllltllllllllllll 190 Xl=O : 12=0 :15=0 :R5=0 200 "=15 :C=50 :L=S :ITER=O :'1 OF DATA USED= "L 210 Olft lift), WIL+l),6125 )1Difti,CI"I,Y1(l+ll 220 FOR 1=0 TO l 230 Wlll=O :Y1(11=0 240 NEIT I 250 6(0)= .s :6111= 1 :6(2)= .8 :6(3)= .2 :6141= .4 260 6(5)= .9 270 LPRINT TA8(30J;"ES" ;TA8142);"1 OF DATA = 10" 280 LPRINT TABIOI;"ITER";TA8(14);"SU" OF RESIDUALS R";TA8(421;"RftS OF R" ;TABI 56l;"STEPSIZE = .01" 290 LPRINT TABIOI;"SIOI";TA8(141;"6(1)" ;TABI2Bl;"GI21";TAB(421;"6131";TABI56) ;"614l";TA81701;"6151" 300 LPRINT TABI 01; 6(0) ;TABI14J; 6111 ;TA8128l; 6121 ;TA81421; 613) ;TABI56 I; 6141 ;TABI70l; 6(5) 310 RETURN 320 RE" lllllllttltlllllll PRINT RESULTS 1111111111111111111111111111 330 CLS 340 LPRINT TA8101; ITER ;TAB!l41; R ;TA8(421; SORIA) 350 LPRINT TAB( 01; WIOl ;TA8(14J; Will ;TABI281; Wl2l ;TA81421; Will ;TABI56 I; W(4) ;TA81701; WISI 360 LOCATE 5 ,1 :PRINT "610)=";610l:LOCATE 6,l:PRINT WIOl 370 LOCATE 5 ,2S:PRINT "6(11=";611l:LOCATE 6,2S:PRINT Will JBO LOCATE 5 ,SO:PRINT "6121=";612J:LOCATE 6150:PRINT Wl21 390 LOCATE 7 ,1 :PRINT "6(l1=";61ll:LOCATE 811 :PRINT Will 400 LOCATE 7 12S:PRINT "6(41=";6(41:LOCATE 8,25:PRINT W(4) 410 LOCATE 7 ,SO:PRINT "6151=";61SI:LOCATE B,SO:PRINT Wl51 420 LOCATE 18,25:PRINT "ITER =";ITER 430 LOCATE 19,25:PRINT "SUft Of RESIDUAlS R=" ;R 440 LOCATE 20,2S:PRINT "RftS RESIDUALS = SORIRI = "; SOAIRI 450 RETURN 102
PAGE 114
460 RE" ltlltlllllllltlltt EXHAUSTIVE SEARCH tllllllllllllltllllllllll 470 60SUB 740 480 Z=2 : 'Z=INT((ITERI/2)+1 490 IF Z<=15 THEN 60TO 510 500 SL=lOA(151 : 60TO 520 510 SL=lOA(ZI 520 FOR N=O TO L 530 60SUB 620 540 NEXT 550 IF AS="O" 60TO 590 560 IF AS="R" 60TO 580 570 ITER=JTER+l : X2=ITER : &OSUB 1200 : GOTO 600 580 ITER=ITER+l : 16=iTER : 60SU8 1430 : GOTO 600 590 ITER=ITER+l : 6DSUB 330 bOO IF ITER= C THEN 60TD 150 :C=IITERATIONS OF EIHAUSTIVE SEARCH b10 &DTO 480 620 REft lllfllllltlllllll SUB PARA"ETERS SEARCH lllllllllllllllltlltll 630 RE" PARA"ETER SEARCH 640 Rl=R:W1=WINI b50 W(NI=W(NJ+Sl:60SUB 740:R2=R:W2=WINI 660 IF R2>Rl THEN SL=Sl:A=R2:R2=R1&Rl=A:A=W2:W2=W1:W1=A:W(NJ=W(NI+SL 670 W(NI=W(NJ+SL:SOSUB 740:R3=R:W3=W(NI 680 PRINT "Rl=";Rl,"R2=";R2,"R3=";Rl 690 IF R1>R2 AND R2>R3 THEN R1=R2 :R2=Rl:W1=W2:M2=WJ:60TO 670 700 W(N)=,51((Wl+W2111RlR2)(W2+WJ)I(R2Rlll/1Rl21R2+Rll 710 PRINT Z,G(NI,M(N),Sl 720 GOSU8 740 730 RETURN 740 REft ltltltllfllllllll CO"PUTE RESIDUALS lllltlttttllllllllllllllll 750 R=O :IPTR=O : 16PTR=L 760 FOR I=L TO "1 770 LPTR=I6PTA 780 Dlfl=O : C(ll=O 790 FOR J=O TO L BOO Dill= Dill+ &IJIIIILPTA) 810 C(fl= Clll+ W(J)IIILPTAI 820 LPTR=LPTR1 :NEIT J BlO 16PTA=I6PTR+1 840 R=R+IDIIIC(I))A2 850 NEXT I 860 R=R/(" LI 870 RETURN 880 'llllltttlttltlllllltlllll INPUTS FOR ES lllllllllllllllttlltttllt 890 RE" Dlkl = G(OIIIIkl+.,, &lllii(KLl 900 FOR K=O TO "1 910 I(K1=21(RND.5)tSUR(li.Oll 920 NEIT K 930 RETURN 940 RE" ttttttltltltttt 1Y PLANE FOR ESTI"ATES lltltlttlttlltltttttll 103
PAGE 115
950 M=1 960 CLS : SCREEN 2 970 FOR Y=O TO 160 STEP 20:LINE (38,Y J(40,Y I : NEIT Y 980 FOR Y=39 TO 589 STEP 1589391/C :LINE IY,79J(Y,811 : NEIT Y 990 LOCATE 11,73 :PRINT C 1000 LOCATE 11,1 : PRINT 0 :LOCATE 1,1 : PRINT M : LOCATE 20,1 :PRINT M 1010 LOCATE 5,1 : PRINT M/2 : LOCATE 16,1 :PRINT M/2 1020 LINE 139,0)(39,160) 1030 LINE (39,80 11589180 1040 FOR 1=0 TO L 1050 1=6111 1060 GOSUB 1090 1070 NEll I lOBO RETURN 1090 IF 1<0 GOTO 1150 1100 Z= 80 (lt199)/(M+M+Mtl9/80) 1110 FOR J=39t1000/600 TO 600 STEP 25 1120 PSET(J,ZI 1130 NEXT J 1140 GOTO 1190 1150 Z=IM+ABSilllt199/(M+M+M139/801 1160 FOR K=3911000/600 TO 600 STEP 25 1170 PSET(K,Zl 1180 NEXT K 1190 RETURN 1200 RE" tltltltllllllllllllt PLOT ESTI"ATES 11111111111111111111111111 1210 M=l 1220 YIEM139,0)(589,160) 1230 MINDOM(O,MIIC,Ml 1240 LOCATE 22,70 : PRINT "ITER=";ITER 1250 FOR 1=0 TO L 1260 LINE ll1,Y110))(l2,W(Oll :LINE 111,Y11111(l2,MIIIl 1270 NEIT I 1280 FOR 1=0 TO L : Y1(1l=M(Il : NEIT I : 11=12 1290 RETURN 1300 &OSUB 1310 ll10 REft llllllttlttlltlll 1Y PLANE FOR RESIDUALS 1111111111111111111 1320 CLS : SCREEN 2 1330 LINE 1119,8 11119,136 I 1340 LINE 1119,136 J(520,136 1350 FOR Y=B TO 136 STEP 8 :LINE 1118,Y I1120,Y I : NEXT Y 1360 FOR Y=119 TO 520 STEP 1520119)/(C11 :LINE (Y,137 )(Y11351: NEIT Y 1370 LOCATE 19,15: PRINT 1 1180 LOCATE 19,65: PRINT C 1390 A= 21 :8= 1 :ftiN=10A(AI : ftAI= lOA(B):l=(ABS(AIABSIBII/16 1400 FOR 1=0 TO 16 STEP 4 :LOCATE 1+2,1 :H=KII+l :PRINT "10E"IHI :NEXT I 1410 FOR 1=0 TO 16 STEP 4 :LOCATE 1+2,11 :H=KII+l :PRINT (H) :NEXT I 1420 RETURN 1430 REn 11111111111111111111111 PLOT RESIDUALS llttlltllllllllllllltl 1M
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1440 A=21 : B=1 1450 LOCATE 2013 :PRINT ;A' :LOCATE 20,9 :PRINT 'LOG!RI' 1460 LOCATE 20,20:PRINT 'SU" OF RESIDUALS'R =';R 1470 LOCATE 21120:PRINT 'R"S OF R = SQR(RI = ';SQR(Rl 1480 LOCATE 22,20: PRINT 'L06(Rl ='; iLD61Rl/L06(101l 1490 LOCATE 23,20:PRINT 'ITER=';ITER 1500 R6=L06(R)/L06(10) 1510 VIEW(l19,B l1520,136) 1520 WINDOW 11, A lIC, B l 1530 IF XS=O GOTO 1SSO 1540 LINEIXS,R5)(16,R6) 1550 15=16 : R5=R6 1560 RETURN 105
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I I I THE }lROSPECTS OF INFORMATION TECHNOLOGY FOR A I FEASIDLE, FGNCTIONING ANARCHOSYNJ?ICALIST DEMOCRACY by Curtis D. Holmes B.A., University of Denver, 1984 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Arts Department of Political Science 1991 :,{'S'"!l ... f\_:." i
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1bis thesis for the Master of Arts degree by Curtis D. Holmes has been approved for the Department of Political Science by ?adde Tecza Date
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Hohnes, Curtis D. (M.A., Political Science) The Prospects Information Technology for a Feasible, Functioning Anarcho Syndicalist Thesis directed by Professor MichaelS. Cummings Many anthropological and political researchers have demonstrated that societies approximating anarchosyndicalist ideal of functional authority without coersion by the state, and of functional economic subgroups where members share radical equality, exist in smallscale, lowtechnological groups. The central focus of this thesis is the prospect of information technologies providing a means to organize a feasible, functioning society approximating the same anarchosyndicalist ideal in mass, developed Caution must be exercised when speculating along these lines by applying social sCience methods to meet some minimum standard of logical coherence and consistency, and by identifying key concepts and lines of inquiry that are central to the constructs of a society. Visions of the "good" political society have I traditionally taken the form of utopian fictional literature. I use the precedent of that I tradition to construct a vision of that society, followed by an analysis of the vision as I related to its theoretical, technological, and anarchosyndicalist characteristics and Throughout each element of analysis, utopian, technological, and I I demonstrate criteria that the political vision meets, and limits that it functions withi4. While too broad to consist of a proof that such a society is feasible and functional, I qonclude that the prospects indicated by analysis within these core areas are There is no implied moral imperative that such a society should arise, only that it possible and desirable under certain conditions and circumstances. The form and content_% abstract are a proved.! recommend its publication. Signed Michael Cummings Ul
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Dedicated to the of Vincent and Olive Holmes.
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CON1ENTS .................................................................... vi CHAPTER 1. INTRODUCTION .............................................................. 1 2. TilE "PROSPECTS" VISION ................................................ 9 3. Tiffi UTOPIAN ANALYSIS ................................................ 27 4. TilE 1ECHNOLOGICAL ANALYSIS .................................... .40 5. TilE ANARCHOSYNDICALIST FOUNDATION ...................... 58 6. CONCLUSION ................................................................ 70 BIBLIOGRAPHY ..................................................................... 72 INDEX .................................................................................. 75 v
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I would like thank the Political Science Department of the University of Colorado at Denver and Of. Michael S. Cummings, Dr. Joel C. Edelstein, Dr. Glenn Morris, and Dr. Thad Tecza in particular for a challenging and inspirational learning experience and dialogue, and alsp for the opportunity to pursue speculative theory. I wish to thank I Kevin W. Perizzplo, Leslie Petrovski, Mary V.G. Lindesmith, and Roxanne Birlauf for their much assistance and patience. I give special thanks to Ron Strube, Esq., for more inspiration and insightfulness than can be expressed. Finally, I wish to thank all .the friends, family, and colleagues who have encouraged and supported the' dissident and somewhat cranky views of a professional malcontent Vl
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"Nihi,l motwn ex antiquo probabile est" : Distrust all innovations, wrote Titus Undoubtedly it would be better were man not compelled to change, qut what! Because he is born ignorant, because he exists only on condition of gradual selfinstruction, must he abjure the light, abdicate his reason, and abandon himself to fortune? Perfect health is better than convalescence. Should the sick man therefore, refuse to be cured? "Reform, reform!'':cried ages since, John the Baptist and Jesus Christ "Reform, reform!" cried our 1father fifty years ago. And for a long time to come we shall shout, Reform, reform! Pierre Joseph Proudhon, Property and Revolution vii
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CHAPTER 1 INTRODUCTION Man the Uto.pia Maker The n,ame of the planet, presuming it has already received one, is immaterial. At its most favorable opposition, it may very well be separated from the earth by only as many miles as there are years between last Friday and the rise of Himalayasa million times the reader's average age. In the telescopic field of one's fancy, through the prism of one's tears, any particularities it presents should be no more striking than those of existing planets. A rosy globe, marbled with dusky blotches, it is one of the countless objec'ts diligently revolving in the infinite and gratuitous awfulness of fluid space. 1958,p. 160) When I first read this initial paragraph of a short story by Vladimir Nabokov, it I struck me like the lost chordthe most perfectly rendered paragraph of English language that I had read. It has continuously transformed its meaning for me, and it strikes me now as a perfect description of the way that people project the world as it should be through their imagination. It is both universally general, a world that approximates our .own enough to be clearly recognizable, and detailed, radically differing to conlni;St the daily "is" with the "ought" Jacob Bronowski in his essay exploring the nature of cultural evolution, The Ascent of Man, discusses, the unique psychological trait of humans of foresight. S.o far, there is nothing to distinguish the athlete from the gazelleall that, in one way or another, is the normal metabolism of an animal in flight But there is a cardinal difference: the runner was not in flight The shot that set him off was the starter's pistol, and what he was experiencing, deliberately, was not fear but .... In themselves, his actions make no practical sense at all; they are ari exercise that is not directed to the present The athlete's mind is fixed of him, building up his skill; and he vaults in imagination into the future. (1973, p. 36)
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From this ability to envision the future, people have contemplated and projected a meliorated politi,cal and economic society for all recorded history. In the Western scholarly tradition, Thomas More gave us the longstanding name, Utopia. "But in Utopia, as later on, we fmd the Thomas More who saw the world as a wicked place and the human heart as a pit of darkness requiring the light of diligent public scrutiny if the monsters lurking there were not to crawl out and devour the person and the society. More began Utopia in a church" (Marius, 1985, pp. 154155). More's genesis of Utopia in a church is prophetic of a long history in Western utopian thought of antiestablishment establishmentarianism. The effon to reform the ills of the world often results in outrage at the state of current institutions because they are not ones that we envisionso the utopians propose new institutions to replace them. I call attention to this aspect of utopian literature, because I would like to distance the thesis from it The I thesis does not assume the necessity of absorbing the individual into some collective will or reformed conununal institution to retain utopian characteristics. It does not, as More does, criticize inunoral or inefficient social institutions by deifying others, but rather measures the power of utopian vision by comparing means and ends and their relationship to one another. The notion of the meliorative perfectibility of people and their society always begs the question of present circumstanceshow did we get here and where to go from here. The utopian: nature of this thesis dwells on the political will of individuals in society to shape ends and means, not on the perfectibility of either. The Role of Information Politics is dependent on conununication, and conununication is limited by a number of factors and characteristics of reality as we know it. Geography and the limits of 2
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people to travel ;rre two of the most important; diversity of language and culture are others. The purpose here is not to identify and itemize a list of limiting factors, but rather to recognize that homo faber, man the maker, by the ability to conceive a technos has continuous!);' remade the nature and scope of those limiting factors. The products of the hand of people, technology, are intimately linked with communication and politics and in many cases provide the imaginative qualities and the limiting reality of utopian political visions. The relationship of technology to politics is often only explored on the way that innovation, assessment, control, production and so on are influenced by the political environment and how political organization (e.g., bureaucracy) can affect technological development. Yet the relationship extends far deeper and wider. It exists at the level of form and control itself. And, in tum, political form determines not simply the efficiency of production of this year's Christmas gift fad, but the existence and of how progress is defmed and the costs that society is willing to pay for the production of goods. In The Control Revolution, James Beniger describes in more detail the specific type of technology that I am interested in, information technology, and its relationship to social control. Because both the activities of information processing and communication are inseparable components of the control function, a society's ability to maintain controlat all levels from interpersonal to international relationswill be proportional to the development of its information technologies. (cited in Teich, 1990, p. 54) One of the ongoing and sharply divided battles in the discussion of issues concerning technology is its inherent nature: good, bad, neutral, all of these? Therefore, in this of computer and information technologies and how they 3
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interact with political structures, the perspective called technostructuralist, elaborated by Tehranian will be used. (1990, pp. 212217) In this viewpoint, technologies are neither good, bad, nor neutral in and of themselves. "This is because they developed out of institutional needs (in the case of infonnation technologies, primarily military and business needs) their impact is always mediated through the institutional arrangements social forces ... (Tehranian, p. 5). In other words, technologies always feed into social and institutional paradigm and have good, bad, and neutral effects. It is the way that the paradigm uses and understands the technologies that will I determine the greatest effect of that technology. Computers, especially when linked to other information technologies, have both centralizing<:ontrol, privileged, barrier ridden and decentralizinguniversal, autonomous, and democratic characteristics. Those characteristics develop in I accordance with the nature of the environment of computer use and the time and place of use. The prorriise of computer infonnation technology is the revitalization of a form of direct democracy and autonomous decisionmaking; the peril, a highly centralized, controlled totalitanan state (with a level of totalitarianism perhaps yet unseen). Prospects also exist for transformation of politics into new forms of association that are less possible or limited by the scale of societies in the absence of computerized information technologies. Tehranian discusses this aspect of computer politics in terms of a Communitaian Democracy. As extensions of our senses and as leverages of power, technologies replicate as well as augment the existing power relations. (Tehranian, 1990, pp. 201212) In its ability to augment, Tehranian discusses the I Green movemen,t in West Germany and the Sarvodaya movement in Sri Lanka as case studies of communitarian value movements or parties whose potential for largescale 4
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reproduction political power has grown and developed along with the information technologies. : I I An important political analysis that must be included in any exploration of current and future political implications of the computer, at the national level in particular, is social choice the.ory, presenting mathematical and logical limits to computational aggregation and :computer politics. "Technology can indeed make democracy possible in places where it was not I possible before. But the possibilities are not boundless. The bounds are set not by technology but by logic" (McLean, 1989, p. 2). Assessment of computer politics is based then on the limits to the very logic systems of which computers are built, but we must asswne new logic systems can and will be created that we cannot yet account for nor anticipate. The AnarchoSyndicalist Moclel The thesis' fmal conceptual element is the foundation of the political theory vision of the social and economic system that I argue can be created given the political will to integrate information technologies into political decisionmaking and to dismantle current institutions. I have chosen as an historical basis the views of anarchosyndicalists with democratic variations. Anarchism speaks to the political questions of property, the state, and the relations between the individual and the state. There was perhaps never an anarchism, but many anarchisms, and they have contributed more in the areas of social critique and political philosophy than in any practical and longlived organizations and associations. Yet practical political application of a philosophy is only one part of its value and not necessary for some form of validity. But for this thesis it is 5
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I a critical requirement that practical application and continued functioning be possibleI that is where I will flesh out the path I think possible. The syndicalist side speaks to the economic questions of ownership, material I wealth, need, supply, production, consumption and the link between the economic I enVironment and the structure of living. Trade unionism was the focus of syndicalist thought historicilly, but I have taken the philosophy to what I consider its logical conclusions, in workerowned and democratically managed finns. Information technology is only an integral part of the operation of such firms but also implies a certain level of P,roduction, technological development, and growth that must be explored to a full picture that answers questions concerning sustainability. There are also two important questions about the parameters of democratically managed firms: (1) What areap:gropriate decisions that society as a whole needs to makethe macroeconomic .questions, and (2) What are appropriate decisions that organizations at the level of the need to makethe microeconomic questions William: Goddwin in Rights of Man and the Principles of Society practically ignores the question of economics and favors a rational analysis based on individual psychology.' This is how he builds his vision of anarchy and radical egalitarianism. He states, "Society is nothing more than an aggregation of individuals" (Horowitz, 1964, p. 113). Yet it is!that aggregatefor which we invented a word, society, that also must be accounted fot and its characteristics included in any feasible theory of anarchism. Pierre Joseph Proudhon analyzes from the perspective of Justice (later also reflected I in the work of Rawls) and states in Property and Revolution, These then are the three fundamental principles of modem society, establishbd one after the other by the movements of 1789 and 1830: 1) Sovert7ignty of the human will, in short, despotism. 2) Inequality of wealth and rank. 3) Propertyabove JUSTICE, always invoked as the guardian angel I 6
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I of sovereigns, nobles, and proprietors; JUSTICE, the general, primitive, of all society. (Horowitz, 1964, p. 106) In anarchism and syndicalism the question is always authority, but the analysis I revolves around ;the nature of the individual and the nature of the aggregate we call society. And, alt:>ng with Proudhon, that analysis must study sovereignty, equality (political and economic), and property. l These then are the three braids, utopian political analysis and theory, information technology, 'and :anarchosyndicalist structure, that will be weaved together here to create a vision, and an analysis of that vision, for a new society. I intend to demonstrate I that such a society is feasible and functional and that information technologies make it possible to sustaJn that vision in a mass, developed, and industrialized society. While all three braids af'e vast areas, I have attempted to focus on the most important aspects I and criteria of ea;ch that applies to the main thesis. I From Ov1d (1st century BC to 1st century AD) through Plato, More, Bellamy, and Orwell, and fron:t utopia to dystopia, political writers have found that a literary expression of visions proves more effective and appropriate than political exposition or methodological social science. The literary style or fonn has a power to I conjure a world whole and communicate that world more directly than might be the case I with methodological exposition. This does not imply that there is a lack of method or valid analysis in expression, only that the style of choice differs substantially from scientific P\lpers. Following that tradition, I present a fictional account of the political and economic information society I see as an option. The method will follow. I The fiction a.t;1d the analysis explicitly treat the individual as primarily a rational actor and the view of tpat individual of the prospects for alternative systems as minimalist. If I the of infonnation technology along the lines explored would in and of itself at a level lead to greater freedom for the individual even without 7
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transforming the society and systems, then it is assumed a sufficient mtionale for following that While developmental ttansformation possibilities exist with information technology and are explored peripherally, those possibilities are not primarily important in the discussion of this thesis. 8
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CHAPTER 2 THE ,'PROSPECTS" VISION: A FICTIONAL JOURNEY The earth turning with its perpetually awakening inhabitants creates dawn as it has been doing and still continues to do; but the people, those groggy, fleabitten, poor, rotten, conscious thinkers, ah, they had to discover that the sun never rises, that the sun does nothing but what the sun does and is no Apollo, but rather a big ball of gas. And Philemon, who prefers Phil, at present a lawyer on this little blue world, has a great deal to do. The trial committee had been formed several months ago by community decree and Phil, contrary to: usual practice, had volunteered to be case presenter. His skills were at least rusty after his years of teaching at the local learning center, yet he couldn't help but be intrigued by the "monster in their midst" as the press so jubilantly termed her. The questions to mind by this case were legion. They propagated, i overpopulated the legal mind and spilled, barren, into the quiet township street Daybreak is ;always the most productive thinking time for him, but today proved to be that irritating exception. He flicked the touchplate on the wall and waited for the room to luminesce. "My Mother," Phil thought, "always warned me about thinking and dressing at the same time. Focus, she always insisted, focus on what you need from the world or it will ignore you my little." He pulled on his typical jumpsuit, a thin grey coverall that had become the dress of the day. Dressing actually proved to be a simple affair I since people had let fashion and its constant change and frivolity pass into history. The one remnant of individuality was the personal vest that most people weaved, dyed or
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painted themselves, or with mother's help, and wore over the jwnpsuit He pulled it on, his thinking idistracted by the pattern. A full crest on the right side of the vest overpowered the remainder of the cloth. He had found this crest, a black and red circle with birds, long ago in a book he had been reading for entertainment. The pattern vibrated before his eyes and captured his imagination .then as it now captured his attention. He had decided then and there to add it to his vest as the powerful symbol of his future life. He didn't need to understand it; its meaning was forgotten anyway; it was the power he sensed in it that he wished to possess for his life. He flicked $e touchplate and left. First on the day's agenda was a visit to the learning center to arrange an indefmite absence. It was fairly routine, especially for case presentation, but, he suspected, the case and his role would become the talk of the community, which constantly led him to question his own motives in this affair. It had been, after all, several years since he last presented, and many would have wanted to argue this case. His almost ridiculous mythological stature in the community guaranteed his admission; however, when he chose to volunteer, he also made many mighty enemies at a stroke. Rounding the adobelike passageway to the center he stopped shortly to adjust his attitude focus on the volley to come. Diana Strand, or S as she was known, would be the difficult and hurt victim in the drama around the comer. She was the center of a fourway marriage in the village and had been manipulating for what seemed decades to make Phil the fi,fth. She was sure to see his mad rush to volunteer for this case as a I desperate escape attempt from her, such was the narrowness of her thoughts. Quickly he tried to concoct some simple formula to try to open her awareness enough so that I she might see that this case was much more to him than everyday life and a simple 10
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appetite for adventure. But formulas wouldn't do. Her life was her marriage and the quadrangle that emerged from it. His concerns, thoughts, worries were beyond her scope; he had to leave her with that at least. "S, I need to talk with youperhaps in private would be best." She remained quite calm yet verbally pounced his intrusion. "Can't you see, I'm teaching class?" S lowered her eyes and walked, shuffling, to the doorway. "I'm sorry; I didn't intend to be so terse, Phil. It's just that we needn't discuss it. I'll talk to you after the trial." The battle not fought, he reflected, can't be lost. *** Phil stepped.into an individual commuter car to drive to an interview and do some research about 60 miles from the village. The car operated on solar energy and had been built to last12 years so far for this one. When he was younger and had little income I credit to use, Phil had entered into an agreement at one of the production factories that manufactured the cars. He worked in the factory for about four months producing the vehicles in order to drive off with one. Even though the cars were not privately owned, I they still represented quite a labor and material commitment in the community and required relatively high amounts of credits to operate. The benefit, however, of considering the complete transportation system as a public goodincluding vehicles had worked to the community's advantage. Rather than a consumable good, vehicles were now planned and produced as a capital good and engineered for extended lifetimes l to provide the mbst transportation and communication for the economic input. The factory experience had been the first time that he had become a member of a firm, even though this was by special temporary arrangement The engineers at this 11
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firm had devised the first operational production system for solarpowered vehicles that needed extraordiparily little battery storage. What storage capacity there was, usually was only neede4 for short periods of time in the worst of weather. Even most overcast days provided enough convertible energy to power the vehicle. His arrival at the plant was made official by a twothirds vote of acceptance by all the workerowners of the plant. Even though this kind of temporary work agreement had become fairly common practice, this factory continued to exercise full voting requirements for ratification of every new workerowner. The vote was taken within the first fifteen minutes of the day when all the policy and daytoday administrative business was performed at computerized information workstations available throughout the plant This realtime, online system allowed all1,200 employees to communicate, argue, and debate during the time limitations, and then to vote. The vote was tallied I almost instantaneously and the next agenda issue, agreed on the day before, became the issue at hand. At the end of all the voting, a summary of all decisions and administrative functions was provided to each person. Although some factories had randomized workerowner roles. within the plant, called facetiously by some, role de jour, this factory had decided on more traditional elections for positions based on merit and testing. Those qualified could run for the various administrative positions of trust and authority. The number of terms, however, were strictly limited to one, and all office holders returned to other positions. The only exception was the engineering departtnent where the head engineer could only be removed by a vote of no confidence. Such was the power of specialized knowledge. I The head engineer in this plant, however, received far lower income credits because of his powerful position. 12
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Phil had extremely happy that this plant was not based on randomized roles as he wouldn't to be distracted by potentially being a candidate during every vote. As a temporary employee, he preferred to work on the car to earn factory credits, rather than in some administrative or specialized position. He did fill in for a sick workerowner for several days in the materials ordering department just before leaving. He left with a mountainous headache daily as his distaste for mathematics fought with his understanding. The ordering computer was interlinked to many of the main materials suppliers of the factory and also to the central ordering computer that any producer could link with. The computer kept constant track of inventory and daily use, attempting to minimize warehousing. Phil's job those few days was to fmd needed materials on the computer market, order, and arrange for transportation. All the income credits and costs in terms of time and value were calculated based on the agricultural supply. Phil mused while ordering a quantity of metal from a factory upriver about his inability to understand how people had seriously placed value, real value, in gold and other metals as carriers of economic worth. That could only happen in an economy abundant with agricultural goods or in such authoritative control that those without fOOd had no avenue of action available. The agricultt,rral supply had been good this year, and the calorie allowance had been going up for several yearssome claimed to ridiculous heights. At the time, calories had become quite the controversy. Agricultural goods and valueadded foods were also available to all on the computer market, but you could only order a maximum calorie equivalent of fo6d That limit had now become so high, that many people ordered the most incredible amounts of chocolate cake and fatadded foods that many were calling for factories to limit production of such goods. 13
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The engineers at this and other vehicle production factories had taken the idea of the I vehicle as a capital social good so seriously and designed these solar commuters so well that orders for the cars had considerably slowed within the last few yearsthus the new temporary agreements arranged to moderate the necessity of decreasing work before people chose to leave. It was, however, work that was reduced and not the workerowners numbers as the bull or the bear within the plant was borne equally by all. The plant hatl no legal status, such as a corporation, but rather operated under the current rules an4 policies decided by the community as a whole. Often this made some operations of the plant more difficult, but preempted the longterm abuse of loopholes in legal semantics by a corrupt few. In fact, many of the factories and work cooperatives changed status and structure often, sometimes evolving into groups that bore little resemblance to their original function or purpose. Longterm stability and purpose provided both a poor means for responding to change and a good environment for developing institutionalized and abusive authority over others. The difficulties inherent in constant change were understood to be a large part of the payment toward a freer life. I His first dayi of work, Phil participated as a full workerowner in setting the agenda and making the decisions of the plants operations. His workstation began with the agenda from the, previous dayan election for a position that had opened after a retirement, a on the reduction of the work day to 5.5 hours caused by dropping demand, a rise automobile pricing to compensate for lost income from the longterm drop in demand,: an arbitration vote on the complaints of the factory manager with the current chief of material acquisition, and a setting of the next day's agenda. 14
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As a student of law at the local learning center, he was very interested in the arbitration vote and how the factory handled such disputes. Apparently, all complaints I had to be made publicly before any remediation is possible. The factory manager had accused the chier of underordering certain materials in the solar electronics deparnnent to make the department's production figures look bad. Apparently, this was a personal vendetta. The charge had been made public at the plant several weeks before and a simple majority of workerowners had voted the charges severe enough to warrant investigation and arbitration. A committee of three randomly chosen workerowners was established to conduct interviews and to review any appropriate documents. They were given three weeks to report. Both reports had been available on the main computer now for one week, and the I arbitration committee's report and recommendation was to be voted on. Phil took this I act very seriously and reviewed the committee's recommendation, the statements of both parties, anq a dissenting view provided by an interested workerowner. It seemed to be fairly clear that the chief had been underordering certain materials that were readily available on the computer market, but was less obvious as to why. The underordering was enough of a concern to others in the plant that the vote clearly favored the report of the committee that irregularities did exist The appropriate response seemed to be removal of the chief, but other options were presented as well, including community service and liquidation of his interest in the plant Moderation won the day and he was simply removed from the role of chief of materials acquisition. Another position at the plant was randomly chosen for him. *** 15
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Phil's wandering thoughts on that first arbitration case at the plant brought his mind screaming back the task before him. His research would take him to a small agricultural and production cooperative about 60 miles outside of town. He was approaching the cooperative and flipped on the map chart just to make sure he had the correct one. He checked the map display cursorily as it was obvious that the building on the right, a farynhouse surrounded by donnitorystyle housing, was the cooperative he was after. He pulled in and sat for a few minutes, building some strength and rehearsing his questions. Severn, the woman who was currently chief negotiator of the cooperative, appeared at the main farmhouse door and waved him in. He had discussed his reasons for coming several weeks ago on the phone but was unprepared for the brute strength and presence she seemed to possess. They greeted traditionally, shaking hands. "Levy is in the main dining room waiting for you," she opened the conversation. "The cooperativ discussed the problem at our last meeting and decided that he should be relieved of duties until the case is completed. He has chosen to stay with us even under these conditions. Be thorough, but don't drag this out" "I appreciate your directness," Phil blurted in return, "my intentions exactly." Phil entered 'the dining room. Levy was very stoic, and volunteered no information, I the perfect reactive personality. "When did you first meet her?" Levy seemed to slightly cross his eyes as he answered, "I don't see that my life is any of your concern." I "You will be asked many questions at the trial," Phil added, "you may as well get some practice in now." 16
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Levy evaluated his position and turned to the window answering every question directly but without embellishment Yes, he had met and admired her. Yes, he knew of her views, probably after the second meeting. Yes, they had met several times at night at the cooperative. No, he was not a member of her group. No, he would not name names. Enough practice. I Phil needed to access some public documents to substantiate members of the cooperative and 'attempt to link others to her meetings in the dark. He asked to use the fannhouse workstation and retrieved names of those he considered interviewing next Severn had made it clear, however, that for now his presence had been tolerated long enough. "Work to do," she stated and opened that front door once again for Phil. On his return trip Phil turned on the car radio and listened as each lowpower transmitter, from cooperatives, some from small villages and towns, some just individuals ranting, faded in and out on the main channels. Many were discussing the upcoming trial, but luckily no one had used his name yet. Fame was rare in an age where everyone's 15 minutes of exposure to fame could stretch on indefinitely thanks to full public media access. One cooperative, as it faded in, appeared to side with many extremists, who, while not yet verbalizing it openly, were discussing the old uses of the death penalty. If you couldn't pick up their notsosubtle implication that it was time to rethink the practice of forbidding death penalties, then you wouldn't realize the I serious implications of this case. As the faded out, the ashram Rama speeded into view and and retreated in the mirror with the temple sign of an avatarlike character holding a handful of onions and smiling somewhat awkwardly With over one million pounds of onions produced last year, the had a lot to smile about 17
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Phil went cfuectly home on his return to the city. He needed time to compile the information he had been researching into something that resembled a presentable case. The case was oqty three weeks away, and the jury had already been selected by computer rote. He flicked the touchplate, and the small cottage began to luminesce from the chemically stored energy from the day's sunlight The cottage was a small, well suited space for one person, and one of about fourteen cottages inhabited by those who taught and worked at the learning center. The cottages were in a wooded area about a mile from the learning center, far enough that Phil didn't feel a8 though he lived in his classroom or at the center, yet close enough to walk or cycle. Originally the learning center cooperative staff had worked and lived in a large mansionsized home they had built themselves, but they found the close proximity of living and working led to frustration and anxiety. Some separation was required. It was then that the community as a whole, including volunteer students, built the cottages as a I living settlement away from the center. It had proven much more effective, though sometimes Phil thought even more distance would be an improvement He scattered his research, interview tapes, press clippings, and assorted artifacts on the floor. *** Her name was Sharon, and it seemed she had always been somewhat precocious. I Her childhood, 'or what was known of it, didn't provide any stunning psychological insights into her later activities, just a story of a very bright young woman with ambition. Her learningcenter activities were average, although the learning center she had chosen to attend, while rigorous academically, had fought several accusations of supporting violent student organizations that had disrupted local town meetings and I work comriluniHes. It had been asked to disband once, but refused, then reformed into 18
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an exclusive organization that accepted no requests for admission. The admissions were by invitation alone. There were no indications that she had ever faced community sanctions, community exile, or a called trial. Apparently while she was carrying on the gravest of her activities, she had also contributed substantially to the communities of which she had become a member and was one of the highest producers on the cooperative where Phil had been conduc;ting interviews. On the surface quite a model. But below that surface .. *** The knock at the door brought Phil screaming back to consciousness. The room spun and shattered into a thousand spinning fragments. The spinning slowed and began to crystallize into a recognizable whole, fragment connecting to fragment until Phil's conscious mind looked out on a unified world that seemed to make some sense. Knock. The door. He stumbled to the door and foundS there, waiting impatiently. "Catching up on a little lost sleep?" she grilled. "Going to invite me in?" Phil only then realized that it was late evening, and he had fallen facedown asleep into the piles of papers detailing the case. "Oh, come on in," he said, "and I'll put on some tea." He assumed that this was to be another assault on his timing for taking this case and another blunt attempt to recruit him into her communal marriage, but his next thought I reminded him iliat she had given up cottage visits for this putpose long ago. "I've come to tell you," she drawled, "that I have been chosen by vote to be the I second case presenter and that we won't be able to have any contact after today." What a relief, thought Phil. "What an honor," he said. "Isn't the community a little behind schedule on appointments?" 19
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"Yes," said S, "but everyone feels that we must be careful on a case of this magnitude. It shouldn't be rushed in any way. We won't be able to speak for quite some time you "Uhhuh," said Phil, "but it will be much easier since we won't be at the learning center until the case is think we'll manage." After she Phil looked at the cottage walls and the papers on the floor. ''Perhaps the world doesn't make whole sense yet," he thought; "need more sleep." *** What makes this inquest significant is that these prisoners represent sinister influences that will lurk in the world long after their bodies have returned to dust They are living symbols of racial hatreds, of terrorism and violence, and of the arrogance and cruelty of power. They are symbols of fierce nationalisms and of militarism, of intrigue and warmaking which have embroiled Europe generatien after generation, crushing its manhood, destroying its homes, and impoverishing its life. They have identified themselves with ... (Jackson, 1947, p. 31) I Phil stood before the court committee, the reporter overseeing the accuracy of the voice computer''s documentation of the case. Phil knew almost every workstation and home would be requesting full transcripts of the case each day. He began by quoting the paragraph from the first international war crimes trial in postWWll Germany as a precedent, even during the time of state domination of politics, addressing the problem they were all about to confront in person. He spoke to the jury, the defendant, and the judge in this case, Jane. She was a member of an agricultural and small motor manufacturing community about 100 miles south: of the city. She apparently had a reputation for having a strong intuitive sense of justice and fairness and had been elected almost uncontested through three I court committees to act as judge in this trial. Her affect was stoic, and it was difficult to detect even the slightest of expressions on her face. Phil had learned long ago when 20
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practicing, however, that the limited role of the judge in modem cases didn't justify the time expense of"guessing the judge." He began carefully presenting all the facts of the case as he had found them. He explained the facts of her childhood, attendance at the academy and the accusations against it, her family background, associations, aspirations. And then, he delved into the crux of the case. Apparently while working on the agricultural cooperative, Sharon had begun forming discussion groups on political and business issues of the cooperatives. Were they producing too much? Was the role of subsistence farming for vegetables at the cooperative useful or a hindrance? But vegetable and aquaculture questions soon turned to authority and culture questions and the group began to lose members because of her tirades against radical equality, but gained others who had heard of her fiery speeches. Phil's interviews at the cooperative had demonstrated consistent stories of her flushed face and rising voice tearing at the fabric of social life at the community and causing serious dissentidns among newly formed groups there. Soon an elite group of her most ardent admirers formed, and when it was discovered they had been hoarding a significant amount of their production, they were asked to leave the community. But this was just the beginning. She wanted to form a core of radical loyalists that she could train to carry out her final solution; she wanted to build an armed group of shock troops who would begin to expropriate land and people for their new society. Those that wouldn't follow, wouldn't live. The anns stash that had been found buried at a small privately leased fann about 20 miles from the cooperative had been designed and built by her and the man everyone considered "second" in their secret society. They had used expropriated equipment and material from manufacturing section of another local commune that produced small 21
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I farm implements and machinery. They had developed molds and had manufactured a variety of shortrange and longrange weaponry to meet the needs of her developing master plan to cqnquer and rule. They had also produced a great deal of ammunition and explosive de;vices through several workerowners at a materials factory 200 miles I south of the city.' "This, above all," stated Phil, "demonstrates the extent of her malevolent irifiltration of our lives and our communities." I Witnesses then were called and facts substantiated. Faces turned in horror and awe as the extent of the threat she had posed and the close proximity of her soontobelaunched authoritarian war became clearerthe "monster in our midst." The judge p<)inted at Phil. "Have you completed presentation?" "Yes," he as she then pointed in the of S. "Second cas presenter." Phil listened as S presented a scenario vastly different from his own. This was relatively especially in cases this well researched, as neither presenter acts as advocate for but rather for the truth and the case committee. Most cases would show one presemer pointing out differences in fact that needed to be resolved or presenting different or more infonnation and interviews that tended to substantiate the validity' or invalidity of the charges against an individual or group. Phil couldn't help but wonder if S wasn't taking this opportunity to humiliate him personally for his marriage reluctarice. S portrayed an ambitious and acrid personality in Sharon, but one that did not deviate at all from the norms of an egalitarian and open society. The small arms, while of enormous concern, could not be directly linked to her except for testimony from witnesses who may be guilty of the offense and attempting to escape I justice. As for her advocacy of a more authoritarian system, belief in such a system had 22
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I long ago ceased to be classified as an offense, and she had been free to create such a I system within the structures of this society as long as participation was strictly voluntary, as PhU's presentation had clearly demonstrated was the case. I S also continued, demonstrating that in fact there was no provable conspiracy, that witnesses to profess that "the monster'' had indeed continued to participate in l both the decisionmaking functions of the f:ums and cooperatives of which she had been a member apd the daily social policymaking functions. How could such a supposed whose beliefs were the antithesis of the egalitarian ideal continue to I participate in a system? And finally, there had been no acts or deeds committed by Sharon or any of her followers that could be considered offenses. Phil thought about S' presentation and realized that even a callous belief that she was acting out of personal revenge discard the points she had raised. He thought she had thrown quite the :into the portrait the press had pieced together of this affair, but little did he know the ;whole case was about to blow wide open. After both ptesentations, the judge pointed to Sharon and asked if she would like to present her and facts of the to the jury. Indeed, she did. She began by I practically admitting all the established facts of the case, but then, in the described fashion, her beet red and the tirade began. "I challenge lfte very validity of this court, this case, these presenters, and any verdict of a strawpicked jury," she railed "By your own beliefs how can you enforce I i such a a verdict. How can you justify the coercive intrusion into my life and welfare, and how will you enforce it except by physical force? Because, opposed to the second presenter, I did construct and bury those weapons; I did form groups who plotted murqer m;td property seizure, and I will continue to build and plot until my last breath. This is no society. For you will have to destroy your society to enslave 23
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me and control actions and words. You will have to coerce and imprison and thus end your ideal, dr I will destroy it for you in a true society of nile by those who merit it; and it will have more wealth than your workerowners, as jittery and gutless as birds who scatter at step, can ever imagine. you will end nonauthoritarianism and the lack of a me, or I will do it for you. "We find your soft Utopias as white I As newcut bread, and dull as life in cells,/ 0 scribes who dare forget how wild we are,/ How human breasts adore alarum bells. I you house us in a hive of prigs and saints I ., Communal, frugal, clean and chaste by law./ I'd rather brood in bloody Elsinore I Or I be Lear's fool, amid the straw./ Promise us all our share in Agincourt I Say that our clerks shall venture scorns and death,/ That future anthills will not be too I good I For Heru)r Fifth, or Hotspur, or Macbeth./ Promise that through tomorrow's spirit war I man's deathless soul will hack and hew its way,/ Each flaunting Caesar climbing to fate I Scorning the utmost steps of yesterday./ Never a shallow jester any more! I Let not Jack Falstaff spill the ale in vain./ Let Touchstone set the fashions for the wise I Ariel wreak his fancies through the rain." (cited in Walters, 1989, p. vii) With Utat,1 a raised fist, and chaos in the court, the day ended. I Phil dichi't bieve that one person who was in that court today, or anyone who read : i the transcripts at :their workstation wasn't mulling about their whole society tonight as he was. He cou14 barely believe the power that Sharon had wielded in that room and the danger that she represented. This was as close to a coup, he supposed, as any society with no state apparatus could come. She had admitted her guilt and challenged the court to find :tier guilty and then control her without destroying its own ideals. Yet it had to be I The court again three days later as the jury had reached a verdict In his meditative state about society, Phil wondered at the continuation of the belief that those 24
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charged with offenses were entitled to face their accusers and the jury in person. It would have beeq. just as easy, if not easier, for the verdict to be polled at workstations and the sentence:carried out by a rote committee with the judge's instructions. But here they were in person, with a hundredthousand people watching computer screens. The jury read the guilty verdict and the judge began the sentencing act with an explanation. "The defendant is found guilty of violating some of the most precious norms of our society. Contrary to her apparent belief, this is not a nonauthoritarian society, but rather one in which authority is decentralized to individuals and to the aggregates that they voluntarily create. Those are the only legitimate modes of authority. But confusing this decentralization with a lack of authority to meet challenges to the continued and freedom of individual choice that you represent is to mistake weakness for strength. Contrary to your argument that we must destroy our most cherished beliefs to physically control you, we protect them. For we have not institutionalized our power and authority as you have done, and have not claimed any elite right to as you have done, but act individually and as a voluntary community to defend ourselves against those who would institutionalize their power and set us on the road to greater inequality and less participation. We make mistakes, but do not then defend our mistakes as a necessary and inescapable part of an institution, but constantly tend toward correction. You have demonstrated that for your own power, and those who follow you, you are willing to kill, injure, destroy, steal, and coerce. You are hereby, banished to the internal urban cooperative where you and others who have been previously banished are free to create your own society as you see fit However, all materials are limited, you cannot leave the cooperative for life, you will be provided with raw agricultural goods and implements only. This is the limit to our authority, an eternal vigilance against you and those who wish to enslave all but a 25
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minority of us as has been done in the past We exercise valid authority to protect ourselves to maintain our society, not to destroy it, by preventing you from destroying it in the name of some selfish glory at the expense of others. 'Do no harm,' is not just a physicians' pledge, but our own. And, 'do not allow harm' is its corollary." *** Phil returned to the cottage and flicked the touchplate as he entered the room. The sun was slipping under the horizon, that big ball of gas that had recharged the chemical walls that now luminesced. He sat at the workstation to order new coveralls and two new pairs of shoes (or his return to the learning center tomorrow. The request almost instantly entered the computer market that presented consumer demand and matched it to current and projected firm supply. A shoe factory 40 miles outside of the city matched the order and entered it into the production planning schedule for the next day. I They also autorriatically ordered enough material from raw producers to meet tomorrow's production demand, which was then picked up by a transportation cooperative and charged back to the material supplier. At ahnost the same moment, a second order for shoes arrived on the demand market from the internal urban cooperative. same process quickly fell into place with a second transportation firm picking up the order for 10 percent less charge than the previous transportation cooperative. The earth tupting with its perpetually drowsy inhabitants creates sunset as it has been doing and still continues to do; but the people, those groggy, fleabitten, poor, rotten, conscious thinkers, ah, they had to discover that the sun never sets, that the sun does nothing but what the sun does and is no Apollo but rather a big ball of gas. 26
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CHAPTER 3 THE UTOPIAN ANALYSIS Is Uto_pia Serious? Chapter 3 presents an exploration of utopia as political theory and its value in social science as a process of model building. It also explicitly analyses the previous fictional section according to certain minimal standards of logic, for example, the requirement of linking means and ends coherently. Before exploring in some detail the social scientific process of elaborating the criteria of utopian political theory qua theory, I will discuss the nature and vaiidity of utopian literature and its place in a particularly political context using More and his Utopia as a primary example. There have been two schools of thought about More's Utopia from his contemporaries to the present day that are generally instructive (Adams, 1975). One school argues that the essential nature of Utopia is that of a joke, at best a satire intended to demonstrate the absurdity of English customs of the period by a "Most Distinguished and Eloquent Author." The other school argues that it is indeed a serious attempt by a man bent on creating a moral political order who presented the essential elements of "The Best State of a Commonwealth" with calculated hwnor. It is difficult, joke or ideal, not to take Utopia seriously at some level from a man who died at the hand of his king ,challenging the morality of political life, if not the whole social order. The good saint demonstrated that lethal seriousness in his advocacy of the burning of heretics as the church attempted to maintain a moral order. Those tWo sqhools extend in scope beyond the controversy of Utopia to all utopias. Is utopia a valid and important construct of our thinking and aspiration, or a bane to
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clear thinking and planning that leaves only utopiaasjoke with any validity? And as a sidebar, there is t,he whole realm of dystopia to consider, of utopia as the epitome of the bad joke. Utopian literature intends to transcend reality; dystopian literature to gaze into the crystal psyche of "manunkind"1 to delimit the fallout from the attempt Generally speaking, three variations can be distinguished in the definitions of the utopia The fJrSt conceives the utopia as a particular literary style and seeks the distinguishing characteristic of it in certain literary qualities. The second the utopia a "utopian," i.e., naive and prescientific, way of thinking about society, for example, in The Development of Socialism from Utopia to Science Engels seeks to distinguish an outdated and prescientific style of SaintSimon, Owen and others from the scientific socialism based in the discovered laws of historic material development The third identifies utopia with the critical approach to the form man has given to society. (Platte!, 1972, pp. 4143) Bronowski in The Ascent of Man, expresses one of the distinguishing qualities of human psychology, and perhaps the sine qua non, as the act of the vaulting athlete whose behavior is driven not by the immediate environment, but by a set of goal directed assumptions lying not in the present but in the future (1973, p. 36). It is a small leap to go this distinguishing quality of human psychology and imagination to the expression of those qualities through oral and written utopian constructs. From this perspective it would be the expectation of a constant and unavoidable expression of l"pity this busy monster,manunkind,/ not. Progress is a comfortable disease: I your victim( death and life safely beyond) I plays with the bigness of his littleness/electrons deify razorblade I into a mountainrange;lenses extend I unwish through curving til unwish I returns on its unself I A world made I is not a world of bornpity poor flesh I and trees,poor stars and stones, but never this I fine specimen of hypermagical/ ultraomnipotence. We doctors know I a hopeless case iflisten:there's a hell/ of a good: universe next door,let's go" "pity this busy monster,manunkind," is reprinted from COMPLETE POEMS, 19131962, byE:. E. Cummings, by permission ofLiveright Publishing Corporation. Copyright 1923,1925,1931,1935,1938,1939,1940,1944,1945,1946,1947, 1948, 1949, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957, 1958, 1959, 1960, 1961, 1962 by the Trustees for the E. E. Cummings Trust. Copyright 1961, 1963, 1968 by Marion Morehouse Cummings. 28
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utopian thinking as an integral part of humanity, rather than utopia as a particular event or story by someone with an agenda. "The critical intention to break through the existing conditions and achieve a better future tum out to be the essence of the utopian phenomenon" (Platte!, 1972, p. 44). The critical view of utopia, however, presents it as a symptom of a simmering crisis, an expression of dissatisfaction with present circumstances and an imaginary vision of different, more amiable ones. And the dystopians warn of the consequences of such imaginative attempts toward change. Others declare that civilization is not an aggregation of the civil individual but the repressive means for survival of death seeking beings. ":According to Freud, civilization is essentially restrictive and repressive. With his pleasure principle man remains fundamentally an enemy of I civilization and its principle of reality" (Platte!, 1972, p. 1 09). The resolution of that contradiction lies in the old trick of moving to a more inclusive category where the contradiction disappears, or at least appears in a greater context Utopia as a critique and of does not exclude it as a normal and constant expression of human psychology toward the future; it is a subset of it And the failure of the scientificsocialist explanation of future reality lies not only in its many failures as a I predictive but also in its a priori assumptions and circular arguments where any historical event c3n fit into the theory to prove it but not to disprove it. or Transformation? So we are left with utopia; utopia that this thesis accepts as serious .. But is it a utopia that vaults:the present into futureoriented, nonexistent circumstances, or a utopia that is a notsosubtle justification for present circumstances, unable to break free of its material and social context? Karl Mannheim (1956) in Ideology and Utopia 29
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analyzes the disqnction between ideology and utopia. This distinction is used I analytically by Kerry Walters (1989) in The Sane Society Ideal in Modern Utopianism. I I Basically, WaltefS argues that opposed to a utopia that breaks the barriers of statusquo I thinking as a prefequisite to social change, ideology, often disguised as utopia, presents a picture of a status quo in a propagandist fashion that justifies the state of current I ideas, classes, artd institutions. Mannheim states, I every period in history has contained ideas transcending the existing order, but these!did not function as [nonideological ideas]; they were rather the I ideologies of this stage of existence.as long as they were "organic4!ly" and harmoniously integrated into the worldview characteristic of the period (i.e., did not offer revolutionary possibilities. As long [for instance] as the and feudally organized medieval order was able to locate its paradise butside of society, in some otherworldly sphere which transcended I history and dulled its revolutionary edge, the idea of paradise was still an I integral part of [the classbound ideology of] medieval society. (1956, p. 193) I "Utopias on other hand, are deliberate attempts to 'transcend' both 'objective' I reality and currently existing ideological structures" (Walters, 1989, p. 67). We have seen, following Mannheim, that utopian thought forms can be from ideological ones on the basis of the difference in their social Ideologies tend to support the conventional, normative and conceptual models operative in a given socioeconomic context. As such, they I tend towards totalization, which in turn social and conceptual innovatiohs and leads to a stagnation which eventually gives rise to alternative utopian Utopias sense the tension between the putatively absolute standards: of ideological structures and actually existent sociomaterial I and strive to alleviate it by introducing alternative worldviews and social In doing so, they chip away at the ideological continuum, thereby as vehicles for social and theoretical innovation. (Walters; 1989, p. 72) I Here we can Walters' thesis that the "sane society ideal" in modem utopian i novels such as /..itoking Backward are ideology, ignoring class realities and, accepting Mannheim's (1956) distinction between ideology and utopia, ask if the previous 30
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chapter's Prospects vision is revolutionary or a redux of technologybased industrial capitalism. The first criticism that could be posed is that the vision of the prospects of info:rmation technology does not change in any fundamental way the nature of mass industrial production and consumption, but only rigidifies it by establishing a paiadox where human liberation, freedoms, and equality totally become dependent on a system of industrial, hightechnology production and innovation, requiring the resultant industrial structure of living/working and work discipline that is by its very nature antithetical to freedom of choice. It could also be argued that such a hightechnology dependent political and economic system will either create a new or sustain an old technocratic elite who will be able to wield an institutionalized form of i authority based on critical, specialized knowledge. This vision is little different than Skinner's (1948)Walden Two where a technocratic meritocracy will rise to power. The difficulty with Walden Two, and many utopian visions is their static quality. It is just that qualicy that is challenged the most by the Prospects vision. This is a dynamic society where no institutionalization of power nor crystallization of roles is legitimate. The latowledge of information technology auhe crux of the social, political, and economic decisionmaking is not of a nature that can be buried in a backyard vault, ' nor a single Yet the dissemination of such knowledge has to be planned, and the nature of temporary and rotating responsibility would make eliteformation because of this specialized knowledge very difficult In view (1956) distinctions, it is also evident that the Prospects vision is not a totalizing vision, but rather one that fragments political and economic society into voluntary aggregates who choose their own operating procedures through democratic means. As for it presenting a basis for a strengthening of bourgeoisie democraticindustrial forms, it is also evident that the basis of bourgeoisie class 31
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exploitation; ownership and property, are devolved to a system of worker' ownership, lease:s, and public/participatory disposition of wealth ari.d power that would I preclude class fopnation and would leave it to critics to explain the functional or material basis of such a class. Means and Ends The Prospeds vision once distinguished as a utopian, not ideological, expression of I I nonnative psychology (answering the question, "what if') can then be subjected to a set I of criteria of utopia as political theory. Harold Rhodes in Utopia in American Political I Thought discusses the "crisis" in American social science at the end of the 1960s and I links it to the crisis of modern utopian dramas. In effect, Rhodes calls for a critical I linking of means and ends in political utopia and one that meets the standards of Robert I Lynd and George Catlin that "Thus if the political scientist is to meet his responsibilities I according to the lyndCatlin criterion, he must (1) perceive a problematic condition, (2) identify an alternative, and (3) specify a method for effecting that alternative" (Rhodes, I 1967, pp. 1011); Rhodes, in attacking the "paradigmatic" view of political thinking outside of scient$c criteria, argues that the responsible social scientist can produce theory that links appropriate means and ends according to methodological criteria. Rhodes argues, and I accept the argument, that "ought" questions can be subjected to I investigation and: empirical verification and that the alternative of an inability of social science is the metaphysical slipperyslope of "ought" questions subject I only to human reason. Or, I would add, in light ofMannheim's criteria, "ought" questions then ideology. Problematic ondition. There are two problematic conditions that can be identified I in the Prospects The first is the political tension between democratic 32
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I participation the growing distance between stateenforced public policy and the i mass While arguments and research continue on the explanation for low voter participation in elections, the basic fact of decreasing participation in the political remains; this at a time when the information available to decisionI makers and to the public as a whole has increased exponentially. And when the speed with which the aggregation of voting has also increased significantly. This condition of the body politic remains a question of whose interests the state serves. In any system dedicated to the proposition that "all men are created equal, can the state remain I a neutral embodiment of that principle, or will it always drift toward serving the interests of one class above others? In his dystopian expose of revolutionary I development, Anj.mal Farm, Orwell proposes that any revolution that establishes an I authority wieldec;l by a minority will grow toward serving the interests of that minority at the expense an increasingly alienated population. If we accept the proposition that the ideal situation is one in which not only "do no harm, but also "do not allow hann, is operative, have we then backed into a minimal state that is justifiable as long as it protects the rights of individuals? Or as Proudhon states,: I To be GOVERNED is to be watched, inspected, spied upon, directed, lawdriven, regulated, enrolled, indoctrinated, preached at, controlled, checked, I estimated, valued, censured, commanded, by creatures who have neither die right nor the wisdom nor the virtue to do so ... .It is, under pretext I of public 'utility, and in the name of the general interest, to be placed under contribution, drilled, fleeced, exploited, monopolized, extorted from, squeezed, hoaxed, then, at the slightest resistance, the first word of complaint, to be repressed ... That is government; that is justice; that is its morality. (N ozick, p. ill) 33
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Whether the utopian vision is consistent with a minimal state or its antithesis is not to be solved but as the crux of the problematic condition of the American polis it is one of the forces behind the Prospects vision. I The second problematic condition is the tension between the political ideology of American society and its economic ideology. It is a question that not only enters at the production end, who owns and who decides, but also at the consumption end, what is I distributive justice, and is there an entitlement to certain economic goods and conditions? This itension between worker and owner, the corporation as a legal individual and individuals as members of corporations, wage labor, the necessity of continued economic growth, supply and demand functions, and so on, lead to I consequences of unemployment, alienation from personal economic wants and needs, powerlessness, poverty, and a totalizing vision of one mode of living and working the discipline of hierarchy, imposed behavior, and over consumption. It is a condition that was explored in two issues of Utne Reader, a bimonthly magazine of articles from the alternative press, in the following titles: "Why Work? When There's So Much More to Life" and ''For Love or Money Making a Living vs. Making a Life." ','The person who works right up to selfdestruction is often accorded far more esteem ihat those seeking a more balanced life" (Moody, 1988, p. 65). "On the other hand9r so they sayyou're free, and if you don't like your job you can pursue happiness by starting a business of your very own, by becoming an 'independenf enhepreneur. But you're only as independent as your credit rating. And to compete in business community, you '11 fmd yourself having to treat othersyour employees' as much like slaves as you can get away with" (Ventura, 1991, p. 78). 34
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This is the central expression of the contradiction between the powerless structure of economic discipijne and the "life" people seek to lead. I an alternative. The Prospects vision identifies anarchism as a positive political theory, j.e., not the negation of the state, but rather the existing civil society before the state its existence. For example, Hobbes "is aware that the state of I I nature in which there is no organized society is a logical fiction; it is the basis for the second fiction, the social contract" (Carter, 1971, p. 15). And I would add that the fiction of the state as a representative of the ''people's will" evolves quickly from the social contract ideal. The difficulty with the anarchism ideal is that for there to be any social decisionmaking there must be an aggregation of individual choices, which then constitutes the "people's will," and at that point we must have some structured system of aggregation and become susceptible to that system's physical and logical limitations. The identified then, is the closest feasible approximation to a functioning stateless Prospects als? identifies a syndicalist type of workerowner agricultural and industrial production as an alternative. This would narrow the contradiction between work and other aspects of living by allowing a wide range of voluntary organizations whose benefits are to be reaped equitably. It also empowers everyone concerned with producti9n and consumption of the products of industry. This does not eliminate difficulties and dissatisfactions, but gives responsibility to all and legitimizes that decisionmaking. Factories will go out of business undoubtedly, but as a result of the aggregated decisions of everyone, rather than a few. This system cannot eliminate risk, only spread its share equitably. The alternatives identified then are to eliminate the state and wield authority through direct public and'frrm decisionmaking, and to devolve property to rights of use, not 35
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ownership, and all workers also owners at the level of the finn. The economic I vision differs from the political because microeconomic decisions need to be made at the microeconomic level. The mass public has a diminishing invesnnent in decisions I made in plants or on farms from which they do not consume nor prosper, and for which they do not work. Method for the alternative. Here is the core of the thesis and of the Prospects utopian vision, that in a mass, developed, industrialized society, information technology provides the means for aggregating the kinds of complex choices needed and implementing them with speed and accuracy, limited only by the logical limits of the aggregating process. The negative of the thesis is simply that in the absence of this technology there 1 are no identifiable feasible and functioning means to achieve the identified to the problematic conditions. This assumes that the I deinstitutionalization and disaggregation of current mass, developed culture is not feasible, as amply by the Chinese cultural revolution that failed not only in its ends, but by its totalizing means became the antithesis of the alternatives presented here. The Place of Technolo&Y What then of technology and its place in utopia? The products of the hand of people, techne, cannot fmally be separated from mind. The division between science and technology the result of utilitarian thinking that science, or knowledge in general, I must have some application, some action, to have value. That analytical utilitarian distinction itself has little value. Techne is always a product of the active mind; mind i and knowledge cannot be severed from their action on and in the world. In this sense, no utopia can a vision without a place for some level of technology, both 36
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social and physiCal. Historically, there have been two main responses to the place of technology in utqpia, those that trumpet the wonder of technology and those that trumpet its oppressiveness. "It is not until the seventeenth century that we find the beginnings of modem exaltation of complex technology" (Sibley, 1971, p. 17). In New Atlantis, Francis Bacon expresses the profound faith of the progressive nature of science and technology and the utopian future that a scientific society would have. This is the major vision of industrial society: as the conqueror of nature to carve out a human rational paradise within irrational nature. It is a totalizing vision of technology as the liberator of the race from original sin and its consequence, irrationality. "Gone is the notion of limits to what man can or ought to do with Nature. The idea of conquering Nature as an army would conquer another nation makes its fullfledged appearance and will have an enormous influence on the subsequent history of men's institutions and conceptions" (Sibley, p. 19). This view is intimately linked with the idea of progress as a linear, additive, upward sloping process of attaining the perfect society. It is also intimately linked with the capitalist industrial revolution and its ideal of unlimited material wealth and unlimited expansion markets. In this genre, technology is not merely the means toward a utopian Vision of society but the embodiment of that utopiatechnology is the good. The otherresponse to technology in utopia has been "[t]he despair of dystopian writers about human capacity to control the technological process it has been initiated [leading] a few utopiasts in the last generation to formulate schemes which either halt technology at rather primitive levels or selectively encourage some types while prohibiting or restricting others" (Sibley, 1971, p. 37). In A. T. Wright's 37
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Islandia, a character defending the Islandian way of life expresses the view: ''Would it be pleasanter if we had come by train from the City?" (Sibley, p. 37). This embodies the antitechnological paradigm presented by many utopian and dystopian visions concerning not only the quality of experience that technology can provide, but the dangers of technology out of control, as in Mary Shelley's Frankenstein. Included in this genre are the dystopias antithetical to technologyasthe good where tec!mologybecomesthebad leading to Orwell's despairing conclusion of I 1984 where the ip.tegrity of the individual no longer exists. Human ambiyalence about technology will remain in any utopian vision and the Prospects vision uses the perspective called technostructuralist, elaborated by Tehranian (Tehranian, 1990, pp. 212217). In this viewpoint, technologies are neither good, bad, nor in and of themselves. ''This is because they developed out of institutional needs (in the case of information technologies, primarily military and business needs) and their impact is always mediated through the institutional arrangements and social forces ... (Tehranian, p. 5). In other words, technologies always feed into :the social and institutional paradigm and have good, bad, and neutral effects. It is the way that the paradigm uses and understands the technologies that will determine the greatest share of effect of that technology. In this view technologies feed into the institutionally created structures, magnifying strengths and weaknesses. The Prospects vision .takes three of the strengths of computerized information technology speed, logic, and memoryto overcome limits imposed by the nature of society and geography. It is the outcome of political will that will determine the characteristics of the technology that prevails. The overall conclusions of the chapter are that utopia, as a serious form of political theory, must conform to certain criteria to have validity. Those criteria include, but are 38
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not limited to, transformative possibilities rather than ideology and coherently linking means and ends. The Prospects vision presents information i technology as the central mechanism for meeting those requirements and I now focus on that 39
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CHAPTER 4 THE TECHNOLOGICAL ANALYSIS The Limits of Political Institutions Chapter 4 looks at the place of information technologies in political structures. Technology has its own internal logic and limits that must be explored both outside of I and in relation to :the political structures it will serve, yet in a way that is nondeterministic. And beyond limits and logic, it also explores technology as a process and its possibilities of meeting the process needs of the Prospects vision. It has been sal.d that economics is the science of scarcity, as all the principles and analyses that spring from the dismal science are based on an assumption of limitations, limited resources, labor, capital, and land. In an analogous way political science is the science of and decisionmaking limitations and the power structures that result There are basic identifiable variables that enter into the portrait of political limitations. One is population of the decisionmaking group. The number of members of a political body will influence other variables, namely, the amount of land needed to produce food, geographical distance between any two individual members and their communities, farr:rily size and structure, ability to assemble and travel, time needed for communication between members and communities, and types of aggregation/counting mechanisms that can be used. All of these variables interact and affect one another on one side of the equation, but on the other side is the resultant political system that must live within the lirtrl.tations of the variables as a whole to be feasible and functioning. One of the most often cited examples of this qualitative equation is the limitation of clirect democracy in mass societies because of geographical and physical constraints.
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I You simply cannot cart off everyone in the society to a mass meeting, and if you could, you wouldn't have a large enough building to house them, nor enough time for universal participation. This basic limitation of reality in economics is referred to as the Production Possibilities Frontier (PPF), basically a zerosum equation with two or more variables. An increase in one, say production of televisions, necessarily leads to decrease in another, artichokes, because capital, labor, and material are limited. Again, we can create an analogous example in political science calling it the Institutional Possibilities Frontier (IPF). You simply cannot create a feasible, functioning institution that requires more personnel, information, or time than it has. One of the of the limits in the PPF and the IPF is technology. A restructuring of capital into a technology that uses the same labor and material more efficiently can raise the PPF and IPF to new limits. It is imponant to point out that limits are not eliminated, but simply redefined at a new level. An economy can perform below the PPF limits, but if maximum production is a valid good, then the situation is not optimal. And again, a political system can also perform below the IPF, but if maximum participation and freedom of the individual are valid goods, then that situation is also not optimal. I Oyercomine Limits: Requirements for Democratic Technology It is the contention of this thesis that while representative democracy may have expressed the outermost limits of the IPF at an historical point in time, that is no longer the case. Technology, specifically, information technology, has pushed that IPF outward and the current political system is functioning suboptimally. Information technologies proVide the necessary means for a mass, developed society to meet, 41
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discuss, and decisions with little regard to geographical, transportation, or population Tehranian (1990) states, At thb risk of oversimplification, however, let me conclude that information technolowes can in fact serve a more democratic world development on at least the follo'Yffig three conditions: First, if they are made more interactive. Second, if they achieve more universality and accessibility. And third, if they are increasingly locked into participatory, democratic institutions and networks. (p. 17) i Some of the first steps toward interactivity have taken place on various scales in several countries. with television and cable. The Qube in Columbus, Ohio, was one such system that provided : interactive responses to viewers who were connected to a main computer at the I I television Not only were preferences recruited directly but also time spent watching television, channel choices, and so on, were recorded by the central computer. The system was instituted in several major cities in slightly different formats. The sysiems offered entertainment, education, public opinion polling, I teleshopping, community interaction. By 1984, however, Warner Amex, owner of Qube, was askink to be released from its agreement under cable regulation. It wasn't the technical of the system but rather a technology quickly out of date and low community participation that led to heavy fmanciallosses. (The Economist, January 28, 1984, p. 27, cited in Tehranian, 1990, p. 127) The Qube seems to have come to ... interactive social dialogue was when in deference to its community obligations [under regulation agreement] it offered channel facilities at no cost for 'town meetings' ... The Commission asked a senes of questions on which the citizens gave their preferences. Results were displayed within seconds after they had pressed the buttons. To make sure of the freedom that comes from anonymity of response, the Qube hosts assured the respondents that the computer had been set in a mode that would noi identify them. Asked if they wanted to do the experiment again, 96 percent pressed the Yes button, and within 10 seconds the computer, having worked out that percentage, relayed it to the home screens. (Tehranian, 1990, p. 127) 42
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The National Science Foundation also sponsored experiments in several U.S. cities to explore possibilities of the mostly unused twoway interactive potentials of cable television (CATV). A project in Reading, Pennsylvania, tested the audio/visual conferencing for providing social service programs to the elderly. A Rockford, lllinois, project applied CATV to inservice training for a single occupational group, firefighters. Anda Spartenburg, South Carolina, project offered formal education to students in their homes. The initial findings were that, at least technically, all the projects were technically feasible and functioned as projected. (Kaiser, Marko & Witte, 1977, pp. 1623) Such experiments have been conducted in many parts of the world, especially in Japan, Europe generally, and (West) Germany. In Berlin, the HeinrichHertzInstitute (HHI) provided interactive cable services to subscribers that included town planning, social services, to authorities, teleeducation, purchases of goods and services, and entertainmeq.t Such a contracted service, available in 1977, proves not only the technical feasibility of such a system, but also its applicability. The mn system is close indeed to the system discussed in the Prospects vision, although the Prospects vision I downplays the entertainment and "television" potentials of such home/workstations. The structure of the system tested relates directly to its function. An interaetive two way cable system feeds into the centralizing character of computer information technologies. Here, users can only interact with the center and receive information from that center. This gives the power of information and control to the center and is only made available from the center. This concept of "dumb" workstations dependent on a i central mainframe computer would not be ideal for creating a world like the Prospects vision. Here, decentralized "smart" workstations that can communicate with a central 43
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I computer, or central computers, and are also capable of pointtopoint I communications :with any other workstation, would be desirable. One of the political applications of computer information technology in the U.S. was in government. "Urban government can be viewed as an information I system in which pata are collected, organized, stored, managed, analyzed, and i retrievedall ultimately for decisionmaking purposes" (Westin, 1971, p. 331). The application of in areas such as urban government in fact created fields of study such as systems science and information management. While many had hoped for a total design and implementation, as with most new technologies, computers were integrated piecemeal and a great deal of information duplication and confusion occurred. I think it is important to stress here that even with about 30 years of integration of technologies in the urban management area, no cities have i yet achieved an integrated and "total" systems operation. It is also important to note that this use of infom1ation technologies so far has only served the bureaucracies and I I officials already institutionalized by urban government; it has not served to open urban I government to the public, nor challenged the basic ways in which U.S. urban centers I do business. If rutything, it has allowed urban centers to increase in population to the size of many snuill countries and still maintain the established order. Some form of I computerized vote tallying has been implemented in almost every local voting precinct, yet again the method of voting has changed little, only the speed and accuracy with I which the has been performed. We are an automated and automating society. I do not think it is at this point to state that the integration of information technologies to date has tended t6 feed into the current institutional political and economic framework by strengthening; rather than challenging it. The most trivial use yet put to computers 44
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has been in where they remain isolated devices with programs to store recipes and addresses or 'produce computerized entertainment But the important question does ' not concern critique but why this this so. Part of the answer lies in the ideologies of technology that are presented "[T]echnophiles tend to be the optimists who believe that the present revolution in information storage, processing, and retrieval has already inaugmated a 'postindustrial, information society' with higher productivity and plenty at the world centers that will eventually trickle down to the peripheries" (fehranian, 1990, p. 4). According to this view, information technologies have already done what they best, by making present systems more.efficient. Returning to Mannheim, even;though most of these portraits of future technosociety couldn't be classified as utopjan, the future they portray has little revolutionary potential. The technologies have deterministic qualities and not surprisingly those qualities tend to be the capitalist/indqstrial and representative democratic ones already in existence. This is ideology. Another perspective that tends to feed into the present institutional framework is the i technoneutrals : .. "who have few theoretical pretensions and considerable interest at stake not to their clients" (Tehranian, 1990, p. 5). These are the consultants and engineers who according to the needs of the institutions they serve. This is also a reflection of the corporatist employee status of most engineers and information processors. The fustitutions they work for do not demand nor desire revolutionary ideas. This "nonideology" is also ideological according to Mannheim's (1956) analysis because it also serves the interests of the classes and structure as they exist. The answer also revolves around economic structures. The Prospects vision society would require an 1enormous capital investment in information technology to meet the second of Tehranian that they be made universal and accessible. The 45
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revolutionary democratic potential in the workplace and society is a damper to most I current to invest in such a large capitalization project They may be putting I themselves out of business and restructuring social and economic power in unpredictable and uncertain ways. Most capital is controlled by corporations or by governmental units through taxation. As discussed above, urban governments have defined the public good as an investment in information technologies that will make their functions easier and more efficient, not a decentralization of power to direct voting. Analogously, corporations have not seen information technologies as a worker empowerment tool, but one that could make them niore efficient The first application of multimillion dollar, roomsized computers that could do little more than handheld calculators today was accounting and payroll. The effk;iency of billing and ordering represented an enormous longterm savings and increased earnings for the corporation. Who will pay? It will have to be I individuals who are invested politically and economically in universal and accessible information technology. The third requirement of Tehranian is that information technologies be "locked into participatory, democratic institutions and networks." Two examples of experiments in the participatory yein are televoting ... a novel method of public opinion polling originally by Vincent Campbell as a new public communication system for the San Jose Unified School District in California" (Tehranian, 1990, p.112). In 1918, Ted Becker, Richard Chadwick, and Christa Slaton of the University of Hawaii revised the San Jose Televote in order to turn it into a scientific irandom public opinion sample of a population. In contrast to conventiQnal public opinion polls, however, the new Televote method attempts to inform the public before sampling their opinions. Following a random selection of Televoters, they were provided with brochures on the issues at stake. Ample time is allowed for the reading of these materials before the 46
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Televote staff calls up the voters to ask for their responses. (Tehranian, 1990, p. 112) A New Zealand televote in 1981 also surveyed a sample population on the future of New Zealand and encouraged televoters to make some difficult choices among four different futures for their country. (Tehranian, 1990) While televoting is another example of the application of information technology, it was basically as another method Qf randomsample polling on certain questions, which of course suffer from the same biases as any survey. Also, the televoting is centertopoint, not point topoint, so that the bias comes directly from those at the center and cannot be overcome by poipttopoint debate and communication. The voter doesn't set the I agenda. Paul Goodman spoke to this question of the place of technology on that Institution Possibilities Frontier discussed above in "Can Technology Be Humane?" I need hardly point out that American society is peculiarly liable to the corruption of inauthenticity, busily producing phony products. It lives by public relations, abstract ideals, front politics, showbusiness communications, mandarin credentials. It is preeminently over technologized. And computer technologists especially suffer for the euphoria of being in a new and rapidly expanding field. It is so astonishing that the robot can do the job at all or seem to do it, that it is easy to blink at the fact that he is doing it badly or isn't really doing quite that job. (Teich, 1990, p. 243) I If the televote is just an extension of opinion polling and the technology is really doing a poor job compared to its potential, then what are the possibilities? Tehranian discusses how ... information technologies, no matter how interactive, cheap, or accessible, do not by themselves lead to democratic formations. The latest [1984] U.S. census data demonstrates the point well. (1990, p. 236) ... the Bureau of the Census has confirmed the suspicions about a widening gap between informationrich and informationpoor. According to this report based on a survey conducted in 1984, some 15 million American adults had computers at home, but only 53 percent used them. Predictably, 'the figures also suggest the creation of a computer elite based on race, sex, and income. 47
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Only about 3.4% of children in households with annual incomes of less than $10,000 bad a computer at home, compared with 37% of children in families with of $50,000. Of those who had them, boys were more likely to use a con;tputer than girls (80% to 66% ), and among adults 63% of men used computers at home compared with only 43% of women. As for race, 17% of all white chijdren used computers at home, compared with 6% and less than 5% of black and Hispanic children, respectively. (The Economist, April23, 1988, cited in Tehranian, 1990, p.156) I In contrast to privately owned or governmentally operated information media systems, Tehranian discusses several characteristics of what he would consider a I communitarian system, all of which I believe are evident in the Prospects vision. They are community ownership and management, deprofessionalization of programming and production, empowerment of audiences, interactive technologies and networks, decentralization, cultural and structural pluralism, and thinking globally, acting locally. The implementatjion of the Prospects vision would not come about because of some I I deterministic quality of information technologies. It would not become a dystopia through those same qualities, but rather would be implemented through the political and economic struggle of those who would want to restructure institutions to take advantage of the speed, logic, and memory capacity of information technologies to expand the IPF to include mass, developed anarchosyndicalist forms. The Logic of the System Having analyzed the place of information technologies in a political context, I now tum to the limits of the speed, logic, and memory capacity of information technologies themselves and the boundaries that creates for the design of institutions. The Baconian i view of progressive science as a limitless enterprise of gathering knowledge to eternally better the human condition was probably most symbolically overturned by the dramatic logic and subsequent experimental verification of Heisenburg's uncertainty principle in 48
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physics. (Heisenburg's uncetainty principle is a mathematical relation that demonstrates that if an locates the exact position of a subatomic particle, the speed cannot be exactly; and conversely that if speed is determined exactly, then location can only be estimated. This is not a limitation of the experimental apparatus, but a limit imposed by nature.) That the very reality that had so cheerl'ully given its secrets to the scientific method would also impose ultimate limits to knowledge pulled the foundation from progressive natural science. In a similar way, Arrow's impossibility theorem within the context of general social choice theory pulled the foundation from under progressive social science which held that once you specified a set of preferred criteria for the good society, only implementation remained. Arrow specified a set of desirable and consistent conditions and discovered, "It is not just that it is difficult to a social choice rule that satisfies all these .... No, it isn'tjust that it is hard; it is impossible" (Kelly, 1988, p. 80). The tension is between the individual and aggregation of individual preferences I (wants and need$) into social decision. This can also be complicated infmitely by unanimous subsets of individuals aggregating subset preferences into society. To maintain some simplicity, I will assume only the tension between the individual and the aggregation of preferences and its application in the Prospects vision. Society then is merely the aggregation of individual preferences into some form of communal action. The crux for the information engineer, according to Arrow's (1970) theorem, is that there is no software program, no technological breakthrough that can overcome the logical limits of the nonexistence of a social choice rule that will simultaneously satisfy all the desirable conditions of a social system. The principle tenet of social choice theory and economics from which it sprang is I the assumption Of a rational actor and rational choice. Certainly many fundamental 49
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social science can be attacked at this foundation level by denying (subject to proof) that iri fact that assumption is untenable. For the purposes of this thesis I don't believe there need be any further exploration of this question. The actor can be rational or irrational as dm the social outputs resulting. My primary concern is with the means of aggregate choice, and that means must always conform to some rational structure. In fact it is more applicable in the realm of information technology and social choice progranuning, because those must be expressed in rational structures that can be processed with technology that conforms to its own rational limits, e.g., at present the limits of binary information coding. The first of the resulting analyses of democratic fonns based on the rational assumption is the contradictions of majority rule. Arrow states, "When I fli'st studied the problem and developed the contradictions in the majority rule system, I was sure that this was no original discovery" (1970, p. 93). In fact, this had been discussed historically by Condorcet, Borda, Laplace, Nanson, Galton, and others. The simplest explanation of this problem is that ... there is not a unique way of extending simple majority voting ,to make decisions among three or more alternatives" (Kelly, 1988, p. 15). For exampl:e, three people voting on three possible alternatives (x, y, z) can end in a situation where their preferences are ordered as the following: 1: xyz 2: yzx 3: zxy In this situation, if we aggregate by Condorcet winning pairs, on the fli'st count x is preferable toy (no. 1 and no. 3 would prefer x toy); on the second county beats z (no. I I 1 and no. 2 prefer y to z); and on the third count z beats x (no. 2 and no. 3 prefer z to x). In other words there is no winner with this vote of preference ordering. This method for aggregating individual preferences among three individuals with three 50
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alternatives can lead to a situation where the logic of the system produces no aggregate decision even each individual is clear in preference. And Arrow's impossibility theorem tells us there is no system with a set of conditions that can aggregate individual preferences into a social choice meeting all the conditions. This doesn't mean that there are no methods for getting around this difficulty in the system, only that the difficulty will always remain. How then are democratic systems to be designed to confront these aggregation limits? A second related question also needs to be asked. What if the system does work and there is a majority victor? What are the consequences for majority rule? If one of the conditions (as it is in Arrow's theorem) is that there be no dictator (and in this thesis more generally, wielder of authority) can the majority act as a dictator? It is a long and well debate4 idea that in direct democracies the majority can treat the minority tyrannically. "As I argued ... the principle of majority rule entails, at the very least, that if one of a set of feasible alternatives is the first choice of a majority of voters, then that alternative ought to be chosen" (Sugden, 1981, p. 176). Majority rule is certainly the key procedural attribute of a democratic system, but it does not follow that where there is majoritY< rule there is democracy. First, it is evident from the contradictions of I majority rule that there are instances where the system simply exceeds its logical limits and cannot produce a social preference aggregate that also corresponds to the preferences of the individuals who voted. Second, there are circumstances where the above rule expressed in Sugden "if there is a majority, the first choice of that majority prevails" as. the aggregate of the individual preferences simply is not the "will" of the majority. And it, is this moral claim that a majority expresses an aggregate social will that gives demoeracy its political force. For example, 51
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A community of 1000 people is made up of two villages, A and B, which are severhl miles apart The question at issue is whether licences should be given to 8llow [pubs] to be set up. A majority of the 600 inhabitants of village A are drinkers; in village B, a majority of the 400 inhabitants are abstainers. Drinkers.are in favour of pubs while abstainers object to the associated noise and traffic. There are four alternatives: that pubs should be licensed in both villages (w), that a pub should be allowed in village A but not in B (x), that a pub should be allowed in village B but not in village A (y), and that no pubs should be allowed in either village (z). The profile of preferences is: 1450: (w,x,y,z); (z,y,x,w); 601750: (w,y,x,z); 7511000: (z,x,y,w). Voters 1;600 live in village A and the rest live in village B. Alternative w, that pubs be allowed in both villages, is the first choice of a majority of the whole community. (Sugden, 1981, p. 177) In this example, a pub, to which a majority of residents of B object for their village, is imposed by a majority of overall aggregated voters. There are also other examples of alternative methods to just simple majority rule, e.g., qualified majorities, logrolling. But I don't want, to stray too far from the main point, which is the question of whether i I the Prospects is simply another form of expanded representative democracy with the majority a tyrannical wielder of power and authority rather than the representative state machinery.' One way out of the morass is the way pointed by what Arrow ( 1970) calls "extended sympathy" and in the context of justice by Rawls. That is essentially to be able to "put yourself in someone else's shoes" or blind institutional roles. How would individuals act, and how would it affect the preference ordering, if they did not know what institutional role they would fill. "It is exemplified, in perhaps an extreme form, by an inscription supposedly found in an English graveyard. 'Here lies Martin Engelbrodde, I Ha'e mercy on my soul, Lord God, I As I would do were I Lord God, I And Thou wert Martin Englebrodde" (Arrow, 1970, p. 114). This is the case in the Prospects vision where the aggregation of individual preferences is tllrough a diverse system of smaller voluntary units whose aggregating I I systems vary, rather than through a totalizing single system of aggregation. For 52
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example, the jury in the Prospects case was chosen by random selection as was the second presenter, the first presenter volunteered based on merit and was accepted by the case committee that the city had formed. The judge was chosen by community election, and the whole process was overseen by all the individuals who had immediate and realtime and also latertime access to the case. The lack of a totalizing system does not guarantee that the limits of aggregation will not appear in some failme to match the aggregate to preference, but it does guarantee that that limitation will not appear because of a totalizing system. The potential for consensus exists because workstations are pointtopoint and preferences can be changed, rather than simply counted. When would consensus be an adequate substitute for authority? An organization whose members have identical interests and identical information wiU be one in which spontaneous consensus would be efficient; each member would correctly perceive the best decision according to his interests, and since the interests are in common, they would all agree on the decision. In faceto face groups, it may be possible to interchange information cheaply enough [italics added] so that the identity of information can be achieved, and if the I group has a sufficiently overriding commonly valued purpose, the identity of interests may also be a valid assumption. (Arrow,l974, pp. 6970)2 The fmal. sanction on majority tyranny, however, is the legitimization of the secession of individuals from any system. While that is true of every political system, usually called reyolution, the cost for secession is usually so high that suboptimal functioning is often preferable. On the other side, concerning the lack of stability and opportunity cost of constant change, there are also simple physical limits to the number of secessions an 'individual can withstand, whether there is an external authority or not. 2 I would extend: this definition of facetoface groups due to the technological expansion of the geography/information limits discussed on page 38. 53
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The Prospects vision also works because the "extended sympathy" requirement can be met by random selection for the shortterm institutional roles necessary for functioning. Why would a person act justly, according to Rawlsian analysis. when it is only a moral question? Make that moral question a reality, however, by randomly choosing roles and the justice is evident (Not that this would guarantee that an individual would act justly in that role, only that this is the best way to make the attempt) Authority and Indiyidua1 Choice I am left with two central questions to explore, the problem of authority in the Prospects vision expressed by the court judgment, and does random authority have any greater moral claim than institutionalized authority. The problem of authority in the I Prospects case expressed by the defendant character, Sharon, "By your own beliefs how can you enforce such a judgment, a verdict? How can you justify the coercive intrusion into my life and welfare and how will you enforce it except by physical force? ... For you will have to destroy your society to enslave me and control my actions and vrords ... You will end nonauthoritarianism and the lack of a state i with me. or I do it for you." Can the court assume a position of authority and enforce its decision for the lifetime of the defendant without destroying the anarchistic society of radical equality it values? Radical equality does not eliminate the human condition of authority arising in any cooperative effort. What it does presume is that authority will be shared and equitable and will not crystalize into status quo institutional forms where individuals attached to certain roles become constant over time. If all individuals have 1the same opportunity and responsibility to wield social authority over time. personal authority and freedom can be maintained while the social authority is 54
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shared equitably. The intrusion into the defendant's life is justifiable because to do so retains the form of power distribution, whereas a lack of intrusion solidifies the absolute authoritative claim of the defendant over that equitable distribution. Does randomly distributed authority have any greater moral claim than institutionalized Smith and Ricardo both demonstrated the superiority of specialization, and how, through trading specialized goods at certain rates of exchange, both parties ende4 up gaining in valued goods. Apply this analysis to the political function of wielding authority. Can two or more individuals be better off by specializing, some in wielding authority, some in yielding it, than both individuals retaining rights td authority? A more technical and complete restatement is useful. The generalized Hobbes argument [that in the absence of authority, there is a "war of each against all," and as a result, "the life of man is poor, nasty, brutish, and short."] presupposes two elements: the superior productivity and complexity of joint production, and the cost of interchanging information. (Arrow, 1974, p 56) In the Prospepts vision the capital investment has been made and the cost of interchanging information lowered to the point that the authority argument is negligible. The question then remams whether the ceding of an individual's (x) authority to another individual or grohp (y) would result in superior productivity and a protection of all individuals greater than if they operated separately. H so, the rational individual would prefer to cede personal authority to others to receive a net benefit. This argument assumes that it is a rational choice that is available. It is also a result of speculation about the arising of unequal authority in the form of institutions and the state. This is the importance of the myth of the social contract. It assumes that because there is unequal authority and the institution of the state exists that rational individuals have signed the contract and found a net benefit in doing so. But where is an 55
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individual to be fomtd who has signed the contract? Where is the person hiding? In fact, we are. all sckialized into the political systems of power and authority without ever being asked to rationally evaluate and "signon." What of an individual who wishes to "signoff' the social contract? The person is no less subject to the political system of power and authority than before and often also loses personal freedom. Thus the state, and systems of authority, as we are socialized into them are coercive. There is no rational choice in fact, only revolution. A rational individual then, it is my contention, would find a system of random, equitable authority where the social contract was a real device and could be "signedon" or "signedoff' to have a higher moral claim than a system of institutionalized authority where no such rational choice is possible. If the institution were noncoercive and allowed individu.als to "signoff," then I would argue that the rational individual would be indifferent to the two systems because individual authority would be retained under either. The noncoercive state is more of a perfected vision of utopia, requiring assumptions of the perfectibility of human beings, than the Prospects vision, requiring no such assumpt:ion. ''The polar alternative to authority would be consensus .... By consensus I any reasonable and acceptable means of aggregating individual interests" (Arrow, 1974, p. 69) . . some of us who have read a little bit of the history of thought have heard of anarchosyndicalism before. Bakunin and Sorel had spoken to the same point many years ago. But it is a real one. There is a demand for what might bel termed sincerity, for a complete unity between the individual and the social roles ... (Arrow, 1974, pp. 1516) I This chapter1has focused on the limits and potential of information technology in the Prospects vision as both transforming and functional. The argument that is presented is that at a minimai level, where no transformation of systems take place but only novel 56
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fonns of process in the same system the Prospects vision still increases the level of individual freedom through noncoercive participation. The question that remains is what model of transformation would best serve maximizing individual freedom with technology as the process means. 57
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CHAPTER 5 THE ANARCHOSYNDICALIST FOUNDATION The Anarchist vision Chapter 5 explores anarchosyndicalism as an appropriate utopian model made functional and feasible through information technology processing. The historical difficulties of the model are presented and questions central to it, authority, property, and noncoercive decision making, are presented. The first difficulty in discussing anarchism in general is that there are many I anarchisms. Like' the many socialisms that were part of the international movement, anarchist has produced collective, radical individual, peasant, pacifist, violent, communist, conspiratorial, and many additional minor versions of theoretical and activist ways to the stateless, cooperative society. Unlike socialism, anarchism has not undergone an analogous open debate and, frankly, open warfare to distill the theory into what is workable or winnable. There are also very few models of anarchist development, of in the real world anarchist societies can confront the daytoday concerns and of a community. This strikes both ways. Anarchism as a critique is powerful because of its theoretical diversity and purity from problems of application. But that same lack of application and historical precedent often gives the I critique little validity. The many anarchisms share the quality that there is no institutional means of coercive enforcement of authority; that does not mean that there is no social order or organization. I want to be qlear on a point. There are numerous examples of anarchist or anarchiststyle communities and cooperative efforts. Michael Taylor in Community,
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Anarchy andLib'eny (1982) discusses anthropologicallybased descriptions of anarchistic communities that include acephalous societies, centralized redistribution systems, and and bigman systems. Acephalous societies (literally, no head) have almost no political or economic specialization and what does exist is ad hoc concentration of authority that flows continuously throughout the membership. I Centralized redistribution systems have crystalized authority around various hereditary or divinely sanctjoned leaderships,.but that authority is under obligation to be used for the benefit of all (usually in the form of provision of large feasts). Nonleaders have more claims on leaders by virtue of their position thari is true of the reverse. Chiefdoms and bigman systems have considerable inequality of prestige and authority which again in large measure, depends on merit and generosity. Taylor includes peasant and intentional communities as examples of anarchistic organization as well. The critical attribute shared by all these systems that Taylor presents is that they have no formal, legitimate means of enforcing what functional authority they may have. That authority is always in. flux: in effect, there is no state. The theoretical exploration of the state arising as in Hobbes' work resorts to speculations on the "state of nature," a state that transparently did not exist, or if it did was extraordinarily short lived because of undesirability of the "war of each against all." This view assumes the continuity of the state form of coerced social welfare from time immemorial and ignores the very process of the state arising from nonstate communities that it purports to explain. "During almost all of the time since Homo sapiens emerged, he has lived in stateless, 'primitive' communities" (Taylor, 1982, p. 33). The Kung! of the Kalihari have in fact maintained such societies up to modern times. The diffi<;:ulty is not in the existence of feasible, functioning anarchist societies but their scale. The crucial question is can this form of organization exist in mass, 59
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developed societies? Authority embodied in the political state and the economic authority of a is the issue surrounding that question. We are communists. But our communism is not that of the authoritarian school: is anarchist communism, communism without government, free communism. It is a synthesis of the two chief aims pursued by humanity since the dawn of historyeconomic freedom and political freedom ... The means of prcx:luction and of satisfaction of all needs of society have been created by the common efforts of all, must be at the disposal of all. (Dolgoff, 197 4, p. 29) The Syndicalist Vision The econotru:c authority is ignored or dealt with cursorily in many of the anarchist perspectives. Some perspectives, for example, peasant anarchism that idealized the life of the European }>easant advocating withdrawal from the state to mutualist agricultural societies, are si111ply inappropriate for exploring the place of anarchist ideas in mass, developed society. For this reason, and because syndicalist organization at the level of the firm can potentially deal with contradictions between microand macroeconomic decisionmaking, anarchosyndicalism makes the most sense as a vision of liberated mass, developed society . . the anarchism practiced and preached by a radical trade unionism proved of a sounder variety. It was frrst and foremost based on the realities of nineteenth century European life; and drew its support from the struggles between classes that was at the center of the historic drama. As one writer put it: "Anarchosyndicalism is par excellence the fighting doctrine of the organized working class, in which the spirit of enterprise and initiative, physical courage and the taste for responsibility have always been highly esteemed." (Horowitz, 1964, p:35) Syndicalism in the form of workerowned and operated frrms is no mere theoretical construct Numerous variations on this theme provide models for approaching empirical questions of how such frrms operate, and how they can be constructed to be 60
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efficient and competitive. Examples of longterm, successful workerowned firms include Plywood Coops in the U.S. Northwest, Mondragon in Spain, the socialist workermanaged industries in Yugoslavia, and many smaller and diverse, yet no less important, firms across the globe (Zwerdling, 1984). But the anarchosyndicalist version of worker agitation and control was less clearly successful. The theoretical expansion of anarchosyndicalism from a narrow classbased tool, "the fighting doctrine .of the organized working class" to a direct economic and political struggle to mankind from the state by Fernand Pelloutier and others, makes of it an appropriate critique of the state and institutionalized authority, not just another analysis of classbased political theory. Yet a renewed use of anarchosyndicalist theory in this thesis reqres some speculation and addressing of the historical reasons for anarchosyndicalist failure. Horowitz in The Anarchists (1964) identifies three problems: 1) They tended to approach socialism as a reality around the comer, rather than a longrang process of social reorganization. (Lack of principles or program.) 2) They abandoned the task of organization. 3) They failed to offer sound sociological or psychological reasons for getting people to act. It failed to distinguish between the ends of action and the stimuli to action. I would add that they also lacked the technical means for organizing the society envisioned in the absence of the state. Engels in his On Authority criticizes the "antiauthoritarians" strongly on the grounds that they assume the new society is about to be born, fullgrown, like Minerva from the head of Zeus. Authority, Engels argues, is just what the new society needs (Marx, Lenin, & Engels, 1974, pp. 100104). What Engels fails to realize in his critique is that the "antiauthoritarians" or anarchists of which he speaks are criticizing the principle of l;!.Uthority on the same basis, namely that a revolution that maintains the state principle of authority gives birth to another state born, fullgrown from the head of 61
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the old. How then can anarchosyndicalism develop a program and principles? The moral imperative is the historic drive toward freedom and equality; it is not rational for an individual to choose less autonomy than human capacity allows, and even if an irrational individual did make that choice, there is no imperative other than coercion preventing a of that choice. The fairness imperative is the opportunity cost of coercion, a cost that has become clear in one arena through the arms race and vast sums of money spent on standing armies across the world. The practical imperative is the continual and longstanding abuses of life and liberty that states act out from political detention to the holocausts of millions. These principles become program when a I means is providect to achieve the anarchosyndicalist vision, a program that includes feasible, functioning workerowned and managed firms, expanded use of direct democratic folll1S (initiative and petition) toward anarchosyndicalist aims, application of technology information gathering and decisionmaking, passive resistance to state authority, and so on. Horowitz's frrst cause of the collapse of anarchosyndicalism historically does not imply that there can be no program or principles as this example I hopefully illustrates. Secondly, the task of organization is one that must occur through the active integration of means to ends. This is Horowitz's weakest point because the anarcho syndicalists did not believe they had to organize because historical development would lead their direction. To say that anarchosydicalists failed to organize because they didn't organize is a bit redundant. Yet, I accept Horowitz's point that the failure was not in their lack,of organization but their belief in the inevitability of historical development Thirdly, there is more than enough "stimuli to action" to analyze since the writings of most anarchists around the time of the French Revolution. The modem technological 62
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state has used coercive and destructive force on scales unimaginable to monarchs and radicals alike. The struggle to constrain state power in the twentieth century has taken an international statist form, however, in the United Nations and other international security organizations, rather than in a restructuring or abolition of the state itself. And, fmally,.the means for mass pointtopoint communication and mass I aggregation of inllividual preferences is available and technically feasible. "A liberated society, I will not want to negate technology precisely because it is liberated and can strike a balance" (Bookchin, 1971, p. 134). I do wish to belittle .the fact that behind a single yard of high quality electric wiring lies a copper mine, the machinery needed to operate it, a plant for producing insulating material, a copper smelting and shaping complex, a transportation system for distributing the wiringand behind each of these complexes other mines, plants, machine shops and so forth .. .let us grant that copper Will fall within the sizeable category of material that can be furnished only be a nationwide system of distribution ... This distribution system need not require the mediation of centralized bureaucratic institutions. (Bookchin, pp. 137138) PrcmertY and Law The mediation necessary in place of centralized bureaucratic institutions that wield coercive authoricy is provided for precisely by the anarchosyndicalist vision. What of property in that vision? The analysis of the origins of property and the origins of the legitimacy of property by many political philosophers is usually rooted in the same "state of nature" fiction as the social contract and the arising state. The difficulty with such an analysis is that the individualist definition of property (as land staked out and mixed with labor for survival resulting in a legitimate property claim) I does not correspond with the world into which we are born. Here there is no unclaimed land, no stakes left to put down, and a system of state sanctioned property rights that 63
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' extend far beyond survival needs. Property is a legal claim on capital, land, and labor that can be obtained, extended, voided, and created by other legal meanswith the state as the coercive authority to back the claim. Proudhon claims in Property and Revolution that is robbery .. Such an author teaches that property is a civil right, born of occupati0n and sanctioned by law; another maintains that it is a natural right originating in labor .. .I contend that neither labor nor occupation, nor law, can create property; that it is an effect without a cause" (Horowitz, 1964, p. 87). He also discusses in The'General Idea of the Revolution in the Nineteenth Century that The people, even those who are Socialists, whatever they may say, want to be and, if I may offer myself as a witness, I can say that, after ten years of careful examination, I fmd the feelings of the masses on this point stronger and more resistant than on any other question. I have succeeded in shaking their opinions, but have made no impression on their sentiments .... that the tpore ground the principles of democracy have gained, the more I have seen the working classes, both in the d.ty and country, interpret these principles favorably to individual ownership. (Proudhon, 1969, p. 210) Property, or ownership, in this version differs from the legal entity by being based on personal use,: rather than speculation. The state simultaneously sanctions certain forms of speculative property claims, when it strengthens its authority interests, and outlaws other fdrms when it threatens those interests. Proudhon s basic conclusion for the use and disposition of property and social goods is by contract rather than law. Rather than tangentially explore the idea of nonstate contracts as legitimate where law I is not and its relation to property ownership, I will simply reflect that the Prospects I vision makes use of this idea of individuals voluntarily entering into community contracts that decide the disposition of property within those communities. That disposition could vary from community to community. It is assumed that under these conditions the djsposition of grandiose speculative property claims would not be in the 64
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best interests of the individuals of the community and would not arise without the authority of the to enforce those claims. The complaint is then often raisedwhat of monopolies? price fixing? cornering the market? Are these not legitimate limitations on the free market that without state authority would develop out of control? Here we must remember that what the state is controlling is its own creation, namely the legal entity of the corporation with recognized rights and duties that the state uses its coercive authority to protect Monopolies and price fixing are the products of the state and its corporate creations, not the free market. I Noncoercive Social Decisions I The "tragedy1 of the commons" probably best describes the "freemarket" concern between public welfare, the commons, and individual rational selfinterest. The argument is set out by Garrett Hardin: A pasture is open to all. The village shepherds keep animals.on the commons and each is assumed to be maximizing his or her own gain. As long as the common pasture capacity can handle the number of grazing animals, a shepherd can add an extra animal without affecting the yield of his animals, others' animals, or the sustainability of the common pasture. However, at that limit the shepherd will have a gain and a loss from adding one animal. However, the net benefit of added milk, meat, etc., accrues to the shepherd while the loss is spread over all the shepherds. In this situation it would be rational for the shepherd to continue adding animals, as net gams would continue to accrue, and the shepherds collectively, all acting in their best self interest, could destroy the ability of the pasture to support livestock at all. Abstracted, this problem becomes a game known as the Prisoners' Dilemma found in Games and Decisions by R. Duncan Luce and Howard Raiffa and reiterated in the 65
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political context by Michael Taylor in Anarchy and Cooperation (1976). Two individuals have a choice of two courses of action. The two individuals must choose their strategies at'the same time, or in complete ignorance of the other's choice. Associated with each pair of choices is a payoff based on four variables all of which are related with decreasing value. Graphically, this would appear as the following matrix: Individual 2 ....1 c D c 'tl X,X z,y .... > D '..5 y,z w,w where y>x>w>zi rows are chosen by individual 1 and columns by 2, and the first entry in each cell of the matrix is the payoff to 1, the second to 2. Each player would obtain a higher payoff if he chooses D (Defect) rather than C (Cooperate), no matter what the other player chooses. However, by choosing D, each player gets a payoff w. A choice I for strategy C would have yielded each player a payoff x, where x>w. We defme a Paretooptimal outcome as one where no other outcome is strictly preferred by at least one player. An outcome that is not Paretooptimal is Paretoinferior. The dilemma is, of course, that the rational choice for maximizing individual welfare (getting payoff y), leads to a Paretoinferior outcome for both players together. Rational individual choice cannot be aggregated into mutual welfare in this game. Even if the individuals could communicate, is no incentive to keep the agreement because defecting from the agreement would lead to a higher individual payoff. Taylor takes :the simple static Prisoners' Dilemma and makes it dynamic by adding the time element and allowing a number of iterations of the simple game. He terms this the Prisoners' Dilenuna Supergame. My purpose in restating Taylor's analysis here is not to reproduce his argument in all its mathematical and logical complexity, but rather 66
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to look at his conclusions based on this analysis. The "tragedy" of the commons and the "Prisoner'' status of the game players is that a rational individual making rational maximizing choices when aggregated with other rational individuals leads to Paretoinferior outcomes. This is certainly one of the basic arguments of modem state theorists for the justification of the state as a welfare maximizer. Hume, for example, argues, Two neighbors may agree to drain a meadow, which they possess in common;' because 'tis easy for them to know each others mind; and each must perceive, that the immediate consequence of his failing in his part, is the abandoning the whole project. But 'tis very difficult, and indeed impossible, that a thousand persons shou 'd agree in any such action; it being difficult for them to concert so complicated a design, and still more difficult for them to execute it; while each seeks a pretext to free himself of the trouble and expence [sic], and wou'd lay the whole burden on others. (Hume, 1888, p. 538) Taylor's of this view involves the results of his Prisoners' Dilemma Supergame a number of strategy outcomes can lead to Paretooptimal results. There are also many strategy vectors where the Prisoner nature of the game remains, and Paretoinferipr results are obtained. The point is that those rational, individual, I maximizing choices aggregated in the supergame can lead to Paretooptimal results. This provides a powerful criticism to the use of such welfare maximizing difficulties in the "tragedy of commons" and "Prisoners' Dilemma" as a justification for the I necessity or imperative of the state. Even assuming a difficulty in maximizing social welfare does not,imply that the coercive state is the only or best solution. To say this, however, is only a critisim of state justifications. These results of the supergame analysis do not provide any positive rationale for the type of anarchosyndicalist society I have presented: The moral claim of the anarchosyndicalist vision lies in the freedom of perpetual individual revolution, of individuals who have the ability to withdraw from any association at a very low comparative opportunity cost relative to that cost when the 67
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state claims sove?egnty. That revolutionary disassociation and voluntary association could be incorporated into a system of social organization that approximates the I analogous free m:az.ket situation in economics. It is the antithesis of Wells' utopia where "[t]he welfare state governed by the samurai sees to it, as to a lesser extent do such states as exist today, that every citizen is 'properly housed, well nourished, and in good health, reasonably clean and clothed healthily"' (Wells, 1967, p. xvii). Wells' totalizing version of the welfare state into the World State sacrifices individual freedoms for rational individual and social maximized happiness. But as Doestoevski's underground man, a version of the radical individual anarchist, proclaims, a human is not a cipher. Human society c8nnot be reduced to the slope of a tangent line on a curve of maximum welfare. Society has borne the tragedy of that heart of darkness many times before. But that tangent line can be used as a basis for individual decisionmaking. That allows an expression of the contradiction of the rational nature of human beings who can fully recognize the meaning of that tangent line and of two plus two equals four, yet choose to put their tongue out at it This the Prospects vision attempts to do and utopian I political theory a).Iows us to contemplate. With9ut a "utopian" commibnent to question the underlying assumptions of social prnctices, proposals for reform tend to bypass .the central problems, and may ameliorate a situation which ought never to be tolerated. Secondly, as Kropotkin indicates ... there is often historical evidence that what seems "utopian" to one generation is accepted as obvious good sense by their successors. Oscar Wilde commented that "A map of the world that does not include Utopia is not worth even glancing at, for it leaves out the one country at which Humanity is always landing ... (The Soul of Man under Socialism, 43). Thiidly, as Kropotkin also emphasized, most people are prisoners of their own and of the reigning conventional wisdom. So their view shuts out large stretches of historical experience, alien areas of social reality, and a vision ot: future possibilities. (Carter, 1971, p. 83) 68
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This chapter has presented the philosophical end behind the Prospects vision. It is a I model that faces historical and logical limitations, but ones that are demonstrated to be within the range of a functional system at some minimal level with information technology as a means. The three braids of analysis of the Prospects vision, utopian political theory, information technology, and anarchosyndicalist theory are complete. 69
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CHAPTER 6 CONCLUSION What I have tried to accomplish is not a proof that information technologies, computers, and their related intercommunication technologies, provide the technical imperative to rationally choose a democratic form of feasible, functioning anarcho.syndicalism. Rather I have attempted to demonstrate the prospects for such a system being feasible and functioning. I am no technocrat or technological pollyanna. While the central and critical proposition of this thesis is that an anarchosyndicalist type of political system not only can be implemented using information technology as a tool to overcome limits of size and information flow, but is preferable because it expands the personal power and responsibility of the individual and the community in alienating cultures and economies, I do not believe that the experience nor research of modem society is that technological fixes, growth as the goal of an economy, nor unceasing technological innovation, will solve the problems of the human condition. I do believe that given the present set of circumstances of social life and organization as the basis for speculation of alternatives, that technologies can be used as integral tools to overcome certain physical and social limits that constrain the possibilities of creating noncoercive and nonauthoritarian systems along the lines of anarchosyndicalist theory. This leaves the question of the malleability of individuals and their psychology and the perfectibility of social life completely out of the picture, or at least at its very edges. Neither of these issues need enter into arguments concerning
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the establishment or functioning of an alternative society whose ends are desirable and whose means for achieving those ends are just, fair, and based on a radical equality. Perhaps redueed population over time, a healed environment, smallscale social, political, and economic organization, and a sustainable lifestyle would liberate human beings in greater quality and depth than any form of mass, technological society, but that is not the question, because that is not where many of the communities of the world exist, nor where most are headed. In that vein what I have discussed is not Utopia, nowhere, but rather Eutopia, a good place, one where all the problems of humanity remain, but in a dynamic and fluid flow of power, and responsibility that does not tend to magnify foibles into the forte of state, institutions, and rigid authority. It is a vision of a society where specialized administrative and complex information is readily available, where an approximated free market exists in real time and where price, supply, demand, and other microeconomic effects respond to real information in real time, and where the cost of information flow and decisionmaking are reduced enough to allow all who wish to participate to do soall with the same technology. It is a vision where society and civilization are aq aggregation of the social and civil nature of individuals and the communities they chose to form and maintain, rather than a set of institutions that require the sacrifice and alienation of untold numbers of people. 71
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BIBLIOGRAPHY I Adams, R. M; (1975). Sir Thomas More, Utopia: A new translation, backgrounds, criticism. N,ew York: W. W. Norton & Company, Inc. Arrow, K. J. Social choice and individual values. (2nd ed.). New Haven: Yale University Press. Arrow, K. J. (1974). The limits of organization. New York: W. W. Norton & Company, Inc. I Benello, C. G., &. Roussopoulos, D. (Eds.). (1971). The case for participatory democracy: Some prospects for a radical society. New York: Grossman Publishers. i I Beniger, J. (1981). The control revolution: Technological and economic origins of the information society. Cambridge, MA: Harvard University Press. Bookchin, M. (1971). Postscarcity anarchism. San Francisco: Ramparts Press. Bronowski, J. (1973). The ascent of man. Boston/Toronto: Little, Brown and Company. Carter, A. (1971).: The political theory ofanarchism. London: Routledge & Kegan Paul. I Cummings, E. E.{1980). Complete poems, 19131962. New York: Harcourt Brace Jovanovich.' Dolgoff, S. (Ed.).'(1974). The anarchist collectives: Workers' selfmanagement in the Spanish revolution, 19361939. New York: Free Life Editions, Inc. I Erasmus, C. J. (1985). In search of the common good: Utopian experiments past and future. NewiYork: Macmillan, Inc. Hexter, J. H. (1965). More's utopia: The biography of an idea. New York: Harper and Row Inc. Horowitz, I. L. (1964). The anarchists. New York: Dell Publishing Co., Inc. Hsu, Fh., Anantharaman, T., Campbell, M., & Nowatzyk, A. (1990, October). A grandmaster chess machine, Scientific American, 263(4), 4450.
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Adams27 }UTow49, 50,51,52,53,55,56 Bacon 37,48 Bakunin56. Becker46 Bellamy? Beniger3 Bookchin 63 Borda 50 Bronowski 1, 28 cable 42, 43 Campbell46 Carter35, 68 Catlin 32 CATV43 Chadwick46 Condorcet 50 decisionmaking.4, 5, 23, 31, 35, 40, 44, 58, 60,' 62, 68, 71 Doestoevski 68 Dolgoff60 Engels 28, 61 Eutopia 71 Freud29 Galton 50 Goodman47 Goodwin 6 Green movement Hardin65 HeinrichHertzInstitute 43 lllii 43 Heisenburg' s uncertainty principle 48 Hobbes 35, 55, 59 Horowitz 6, 7, 60, 61, 62 Hume67 Institutional Possibilities Frontier 41 IPF 41, 48 Kaiser43 Kelly 49, 50 Kropotkin 68 Laplace 50 Luce65 Lynd32 Mannheim 31, 45, INDEX 75 Marius 2 More2, 7 Utopia 2, 27 Nabokov 1 Nanson 50 Nozick 33 Orwell7, 38 dystopian 33 Ovid7 Pelloutier 61 Plato7 Plattel28 Production Possibilities Frontier 41 PPF41 Proudhon vii, 6, 33, 64 Qube42 Raiffa65 Rawls 6, 52, 54 Rhodes 32 Utopia 32. Ricardo 55 Sarvodaya movement 4 Shelley 38 Sibley 37, 38 Skinner 31 Slaton46 Smith 55 social choice theory 5, 49 Sorel 56 Sugden 51, 52 Taylor 58, 59, 66, 67 Tehranian 4, 38, 42, 45, 46, 47, 48 Teich 3, 47 teleeducation 43 teleshopping 42 Televote 46, 47 televoting 46,47 Trade unionism 6, 60 Walters 30 Wells.68 Utopia 68 Westin44 Wright37
